Elements of Continuum Biomechanics

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.

• Dedication ix Preface xiPart One A one-dimensional context 11 Material bodies and kinematics 31.1 Introduction 31.2 Continuous vs. discrete 61.3 Configurations and deformations 91.4 The deformation gradient 141.5 Change of reference configuration 151.6 Strain 161.7 Displacement 181.8 Motion 191.9 The Lagrangian and Eulerian representations of fields 221.10 The material derivative 241.11 The rate of deformation 261.12 The cross section 272 Balance laws 292.1 Introduction 292.2 The generic Lagrangian balance equation 302.2.1 Extensive properties 302.2.2 The balance equation 312.3 The generic Eulerian balance equation 352.4 Case study: Blood flow as a traffic problem 372.5 Case study: Diffusion of a pollutant 392.5.1 Derivation of the diffusion equation 392.5.2 A discrete diffusion model 412.6 The thermo-mechanical balance laws 422.6.1 Conservation of mass 422.6.2 Balance of (linear) momentum 432.6.3 The concept of stress 442.7 Case study: Vibration of air in the ear canal 452.8 Kinetic energy 502.9 The thermodynamical balance laws 552.9.1 Introduction 552.9.2 Balance of energy 562.9.3 The entropy inequality 582.10 Summary of balance equations 592.11 Case study: Bioheat transfer and malignant hyperthermia 613 Constitutive equations 693.1 Introduction 693.2 The principle of determinism 703.3 The principle of equipresence 723.4 The principle of material frame-indifference 723.5 The principle of dissipation 753.6 Case study: Memory aspects of striated muscle 793.7 Case study: The thermo(visco)elastic effect in skeletal muscle 853.8 The theory of materials with fading memory 903.8.1 Groundwork 903.8.2 Fading memory 933.8.3 Stress relaxation 953.8.4 Finite linear viscoelasticity 964 Mixture theory 994.1 Introduction 994.2 The basic tenets of mixture theory 994.3 Mass balance 1014.4 Balance of linear momentum 1024.4.1 Constituent balances 1024.4.2 Mixture balance 1034.5 Case study: Confined compression of articular cartilage 1064.5.1 Introduction 1064.5.2 Empirical facts 1074.5.3 Field equations 1084.5.4 Nonlinear creep 1124.5.5 Hysteresis 1154.5.6 The linearized theory 1154.6 Energy balance 1214.6.1 Constituent balances 1214.6.2 Mixture balance 1234.7 The entropy inequality 1244.8 Chemical aspects 1254.8.1 Stoichiometry 1254.8.2 Thermodynamics of homogeneous systems 1294.8.3 Enthalpy and heats of reaction 1314.8.4 The meaning of the Helmholtz free energy 1344.8.5 Homogeneous mixtures 1354.8.6 Equilibrium and stability 1374.8.7 The Gibbs free energy as a Legendre transformation 1384.9 Ideal mixtures 1404.9.1 The ideal gas paradigm 1404.9.2 Mixtures of ideal gases 1414.9.3 Other ideal mixtures 1454.10 Case study: Bone as a chemically reacting mixture 145Part Two Toward three spatial dimensions 1515 Geometry and kinematics 1535.1 Introduction 1535.2 Vectors and tensors 1535.2.1 Why Linear Algebra? 1535.2.2 Vector spaces 1555.2.3 Linear independence and dimension 1565.2.4 Linear operators, tensors, matrices 1585.2.5 Inner-product spaces 1615.2.6 The reciprocal basis 1625.3 Geometry of classical space-time 1645.3.1 A shortcut 1645.3.2 R3 as a vector space 1655.3.3 E3 as an affine space 1665.3.4 Frames 1665.3.5 Space-time and observers 1695.3.6 Fields and the divergence theorem 1705.4 Eigenvalues and eigenvectors 1765.4.1 General concepts 1765.4.2 More on principal invariants 1785.4.3 The symmetric case 1805.4.4 Functions of symmetric matrices 1825.5 Kinematics 1835.5.1 Material bodies 1835.5.2 Configurations, deformations, motions 1835.5.3 The deformation gradient 1855.5.4 Local configurations 1875.5.5 A word on notation 1875.5.6 Decomposition of the deformation gradient 1885.5.7 Measures of strain 1935.5.8 The displacement field and its gradient 1945.5.9 The geometrically linearized theory 1965.5.10 Volume and area 1985.5.11 The material derivative 2015.5.12 Change of reference configuration 2035.5.13 The velocity gradient 2046 Balance laws and constitutive equations 2076.1 Preliminary notions 2076.1.1 Extensive properties 2076.1.2 Transport theorem 2086.2 Balance equations 2106.2.1 The general balance equation 2106.2.2 The balance equations of Continuum Mechanics 2146.3 Constitutive theory 2236.3.1 Introduction and scope 2236.3.2 The principle of material frame-indifference and its applications 2246.3.3 The principle of thermodynamic consistency and its applications 2286.4 Material symmetries 2316.4.1 Symmetries and groups 2316.4.2 The material symmetry group 2326.5 Case study: The elasticity of soft tissue 2366.5.1 Introduction 2366.5.2 Elasticity and hyperelasticity 2366.5.3 Incompressibility 2386.5.4 Isotropy 2416.5.5 Examples 2426.6 Remarks on initial and boundary-value problems 2487 Remodelling, aging, growth 2557.1 Introduction 2557.2 Discrete and semi-discrete models 2627.2.1 Challenges 2627.2.2 Cellular automata in tumour growth 2647.2.3 A direct model of bone remodelling 2667.3 The continuum approach 2687.3.1 Introduction 2687.3.2 The balance equations of volumetric growth and remodelling 2697.4 Case study: tumour growth 2737.5 Case study: Adaptive elasticity of bone 2777.5.1 The isothermal quasi-static case 2817.6 Anelasticity 2827.6.1 Introduction 2827.6.2 The notion of material isomorphism 2837.6.3 Non-uniqueness of material isomorphisms 2867.6.4 Uniformity and homogeneity 2877.6.5 Anelastic response 2897.6.6 Anelastic evolution 2907.6.7 The Eshelby stress 2967.7 Case study: Exercise and growth 3017.7.1 Introduction 3017.7.2 Checking the proposed evolution law 3017.7.3 A numerical example 3037.8 Case study: Bone remodelling and Wolff’s law 3058 Principles of the Finite Element Method 3098.1 Introductory remarks 3098.2 Discretization procedures 3108.2.1 Brief review of the method of finite differences 3108.2.2 Non-traditional methods 3138.3 The Calculus of Variations 3138.3.1 Introduction 3138.3.2 The simplest problem of the Calculus of Variations 3158.3.3 The case of several unknown functions 3218.3.4 Essential and natural boundary conditions 3238.3.5 The case of higher derivatives 3268.3.6 Variational problems with more than one independent variable 3298.4 Rayleigh, Ritz, Galerkin 3308.4.1 Introduction 3308.4.2 The method of Rayleigh and Ritz 3328.4.3 The methods of weighted residuals 3348.4.4 Approximating differential equations by Galerkin’s method 3368.5 The finite element idea 3418.5.1 Introduction 3418.5.2 A piecewise linear basis 3438.5.3 Automating the procedure 3488.6 The FEM in Solid Mechanics 3538.6.1 The Principle of Virtual Work 3538.6.2 The principle of stationary potential energy 3588.7 Finite element implementation 3598.7.1 General considerations 3598.7.2 An ideal element 3608.7.3 Meshing, insertion maps and the isoparametric idea 3628.7.4 The contractibility condition and its consequences 3638.7.5 The element IVW routine 3668.7.6 The element EVW routine 3688.7.7 Assembly and solution 369