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Biomechanical Aspects of Soft Tissues

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Biomechanical Aspects of Soft Tissues Biomechanical aspects of soft tissues Benjamin Loret Université de Grenoble, France Fernando M.F. Simões Instituto Superior Técnico Universidade de Lisboa, Portugal November 19, 2016 Crée comme Dieu Ordonne comme un roi Travaille comme un esclave Constantin Brancusi Contents Page Contents 4 1 Biomechanical topics in soft tissues 23 1.1 Introduction.................................... ........ 23 1.2 Diffusion, convection, osmosis: towards wearable artificialkidneys. 24 1.3 Issuesindrugdelivery . .. .. .. .. .. .. .. .. .. .. .. .......... 25 1.4 Macroscopic models of tissues, interstitium and membranes ................. 27 1.5 Energy couplings, passive and active transports . .................. 28 1.6 Moregeneraldirectand reversecouplings . ............... 29 1.7 Tissue engineering re-directed to tumor tissue exploration .................. 29 1.8 Frommechanicstobiomechanics . ........... 30 1.9 Mechanisms of injury of the knee, fracture mechanics . .................. 31 1.10 Water and solid constituents of soft tissues . ................. 33 I Solids and multi-species mixtures as open systems : a continuum mechanics perspective 35 2 Elements of continuum mechanics 37 2.1 Algebraic relations and algebraic operators . ................. 38 2.1.1 Notations and conventions for tensor calculus . ............... 38 2.1.2 Algebraicoperators . .. .. .. .. .. .. .. .. .. .. .. .. ....... 39 2.2 Characteristic polynomial, eigenvalues and eigenvectors ................... 42 2.2.1 Eigenvaluesandeigenvectors . .......... 42 2.2.2 Characteristic polynomial of a second order tensor . ................ 43 2.2.3 Scalarinvariantsof the stress deviator . .............. 44 2.2.4 Spectral analysis of a symmetrized dyadic product of twovectors . 46 2.3 Afewusefultensorialrelations . ............. 47 2.3.1 Cayley-Hamiltontheorem . ........ 47 2.3.2 Application to the polar decomposition F = Q · U .................. 47 2.3.3 GeneralizedCayley-Hamiltonrelation . ............. 48 2.3.4 The tensor equation X · A + A · X =2 S ....................... 48 2.3.5 Rank one and rank two modifications of the identity . ............. 49 2.3.6 Square-root of a positive definite symmetric second ordertensor . 50 2.4 The differential operators of continuum mechanics . ................. 51 2.4.1 Definitions of the differential operators . .............. 51 2.4.2 Applicationtokinematics . ......... 55 2.4.3 Spherical, cylindrical and polar coordinates . ................ 56 2.4.4 Convectiveacceleration . ......... 61 2.5 Measuresofstrain ................................ ........ 63 2.5.1 Polar decomposition of the deformation gradient . ............... 63 2.5.2 Spectral decomposition of the right and left Cauchy-Greentensors . 63 2.5.3 Lagrangianand Eulerian strain measures . ............ 64 4 2.5.4 Rate of deformation and rates of strain measures . .............. 64 2.5.5 Representationtheorems for rotations . ............. 65 2.5.6 Infinitesimalstrains . ........ 69 2.5.7 Volume change and incompressibility . ............ 70 2.6 Transports between reference and current configurations................... 70 2.7 Time derivatives and Reynolds theorems . .............. 73 2.8 Work-conjugatestress-strainpairs . ............... 75 2.8.1 Stress conjugate to a given Lagrangian strain measure ................ 75 2.8.2 Issuesrelated to the logarithmicstrain . .............. 76 2.9 Smalluponlarge .................................. ....... 77 2.10 Kinematical constraints and reaction stresses . ................... 78 2.11 Invariance,objectivity,isotropy . ................. 79 2.11.1 Tensorinvariance. ........ 80 2.11.2 Material frame indifference or material objectivity .................. 80 2.11.3 Materialsymmetries . ........ 84 Exercises on Chapter 2 .................................... 90 3 Thermodynamic properties of fluids 95 3.1 Basicthermodynamicdefinitions . ............ 95 3.1.1 Open and closed systems. Intensive and extensive properties ............ 95 3.1.2 Energy, free energy, enthalpy, free enthalpy, Legendreduals ............. 96 3.1.3 Chemical equilibrium between two multi-species phases ............... 97 3.1.4 TheGibbs-Duhemrelation . ....... 98 3.1.5 Chemical affinity associated with a chemical reaction . ................ 99 3.1.6 Chemical equilibrium versus steady state . ............. 100 3.1.7 Flux- and energy-driven chemical reactions . .............. 100 3.2 Phasechange ..................................... ...... 102 3.2.1 Phasechangeand the Clapeyronformulas . ........... 102 3.2.2 The principle of Spite and the Kelvin-Helmholtz formula............... 105 3.2.3 Thetimecourseofadesiccationprocess . ............ 107 3.2.4 Negative pressures, metastability and cavitation . ................. 108 3.3 Thermodynamic functions of fluids . ............ 109 3.3.1 Theperfectoridealgas . ....... 110 3.3.2 Compressible and incompressible liquids . .............. 111 3.3.3 Explicit thermodynamic functions for liquids . ............... 112 3.3.4 Liquid-vapor equilibrium curve: IAPWS and simplified formulations . 118 3.4 Laplace’s law of mechanical equilibrium at the interface between two immiscible fluids . 123 3.5 Henry’s law and Raoult’s law of chemical equilibrium betweenaliquidandagas . 124 3.6 Composition of fluids in plasma, interstitium, cell . ................... 125 4 Multi-species mixtures as thermodynamically open systems 129 4.1 Thethermodynamicallyopensystem . ............ 130 4.2 Chemomechanical behavior and growth of soft tissues . ................. 130 4.3 Generalformofbalanceequations . ............ 134 4.3.1 Partial, apparent and intrinsic properties . ............... 134 4.3.2 Mass, geometry and kinematical descriptors . .............. 134 4.3.3 Integralandlocalbalanceequations . ............ 135 4.4 Balanceofmass................................... ....... 137 4.4.1 Balanceofmassforspecies . ........ 137 4.4.2 Balanceofmassforthemixture . ........ 138 4.4.3 Diffusion,masstransferandgrowth . .......... 138 4.5 Balanceofmomentum ............................... ....... 139 4.5.1 Balanceofmomentumforspecies. ......... 139 4.5.2 Balanceofmomentumforthemixture . ......... 140 4.5.3 Interactionmomenta. ....... 141 4.5.4 Interactionstresses. ......... 141 4.6 Balanceofenergy................................. ........ 142 4.6.1 Notation of thermodynamic functions in a mixture context ............. 142 4.6.2 First principle of thermodynamics . ........... 143 4.6.3 Balanceofenergyforspecies . ......... 143 4.6.4 Balance of energy for the mixture: direct derivation . ................ 144 4.6.5 Thecaseofanelectrolyte . ........ 146 4.7 Balance of entropy and Clausius-Duhem inequality . ................. 146 4.7.1 Balanceofentropyforspecies. .......... 147 4.7.2 Internal and external entropy productions for a species................ 147 4.7.3 Second principle of thermodynamics . ........... 149 4.7.4 TheClausius-Duheminequality . .......... 150 4.8 Dissipationmechanisms . .......... 153 4.8.1 Dissipation due to generalized diffusion and transfer ................. 154 4.8.2 Spontaneous reactions and stable configurations . ............... 154 4.8.3 Coupledchemicalreactions . ......... 156 4.8.4 Dissipation associated with a chemical reaction . ................ 157 4.8.5 Are growth and structuration dissipative processes? ................. 158 4.8.6 Onsagerrelationsandratepotential . ............ 158 4.9 General constitutive principles . .............. 159 Appendices of Chapter 4 .................................. 161 5 Anisotropic and conewise elasticity 165 5.1 Hypoelasticity, elasticity and hyperelasticity . ..................... 166 5.1.1 Ahypoelasticmaterial. ........ 167 5.1.2 Anelasticmaterial.. .. .. .. .. .. .. .. .. .. .. .. .. ....... 168 5.1.3 Aparticularhyperelasticmaterial . ............ 168 5.2 Anisotropiclinearelasticity . .............. 169 5.2.1 Matrixrepresentation . ........ 169 5.2.2 Minorandmajorsymmetries . ....... 170 5.2.3 Microstructureandsymmetrygroups. ........... 170 5.2.4 From anisotropic to isotropic linear hyperelasticity.................. 173 5.2.5 Linearisotropicelasticmaterials . ............. 176 5.2.6 Linearorthotropicelastic materials . .............. 177 5.2.7 Prestress/prestrain in linear hyperelastic materials.................. 180 5.2.8 Linear transversely isotropic elastic materials . .................. 180 5.2.9 Linearcubicelasticity . ......... 183 5.2.10 Saint-Venant’s relations for linear elasticity . ................... 184 5.2.11 A particular elastic material with fabric anisotropy.................. 184 5.3 Spectralanalysisoftheelasticmatrix . ............... 186 5.3.1 Spectral analysis: linear elastic isotropic materials .................. 187 5.3.2 Spectral analysis: linear elastic transversely isotropicmaterials . 187 5.3.3 Spectral analysis: linear elastic orthotropic materials................. 188 5.4 Fiber-reinforced materials: extension-contraction response .................. 189 5.4.1 Basic remarks on the mechanical response of fiber-reinforcedtissues . 189 5.4.2 Conewise constitutive response: generalities . ................. 190 5.4.3 Application to materials with one family of fibers . ............... 191 5.4.4 Application to materials with two families of fibers . ................ 191 5.4.5 Application to linear orthotropic materials: octantwiseresponse . 192 5.4.6 Application to transversely isotropic materials . ................. 194 5.4.7 Application to isotropic materials . ............. 194 5.4.8 Cartilageasa tissuewith cubic symmetry . ............ 194 5.4.9 Cartilageasanorthotropictissue . ............ 194 Exercises
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