Multiscale Biomechanics provides new insights on multiscale static and dynamic behavior of both soft and hard biological tissues, including bone, the intervertebral disk, biological membranes and tendons. The physiological aspects of bones and biological membranes are introduced, along with micromechanical models used to compute mechanical response. A modern account of continuum mechanics of growth and remodeling, generalized continuum models to capture internal lengths scales, and dedicated homogenization methods are provided to help the reader with the necessary theoretical foundations. Topics discussed include multiscale methods for fibrous media based on discrete homogenization, generalized continua constitutive models for bone, and a presentation of recent theoretical and numerical advances.
In addition, a refresher on continuum mechanics and more advanced background related to differential geometry, configurational mechanics, mechanics of growth, thermodynamics of open systems and homogenization methods is given in separate chapters. Numerical aspects are treated in detail, and simulations are presented to illustrate models.
This book is intended for graduate students and researchers in biomechanics interested in the latest research developments, as well as those who wish to gain insight into the field of biomechanics.
1. Introduction - p. xiii 2. Part 1: Theoretical Basis: Continuum Mechanics, Homogenization Methods, Thermodynamics of Growing Solid Bodies - p. xxi 3. 1: Tensor Calculus - p. 3 4. Abstract - p. 3 5. 1.1 A short historical vignette - p. 3 6. 1.2 Vector spaces - p. 5 7. 1.3 Covariant and contravariant tensors - p. 7 8. 1.4 Linear forms and duality - p. 8 9. 1.5 Tensor algebra - p. 11 10. 1.6 Euclidean tensors - p. 17 11. 1.7 Algebraic operations on tensors - p. 27 12. 1.8 Differential calculus on tensors: tensor analysis - p. 47 13. 1.9 Differential operators in curvilinear coordinates - p. 53 14. 1.10 Partial derivatives of function with tensor arguments - p. 65 15. 1.11 Elements of functional analysis - p. 71 16. 2: Continuum Mechanics - p. 75 17. Abstract - p. 75 18. 2.1 Motivations of nonlinear mechanics1 - p. 75 19. 2.2 Prerequisite: summary of linear elasticity - p. 83 20. 2.3 Introduction: notion of body in a continuum description - p. 85 21. 2.4 Kinematics: displacement, transformation gradient, strains - p. 86 22. 2.5 Deformation tensors - p. 100 23. 2.6 Polar decomposition theorem - p. 107 24. 2.7 Linearization of the kinematics: small strains and small displacements - p. 110 25. 2.8 Deformation velocities - p. 112 26. 2.9 Transport operations: pull-back and push-forward - p. 114 27. 2.10 Isotropic tensor functions - p. 118 28. 2.11 Stress measures and strain–strain duality - p. 121 29. 2.12 Balance laws - p. 137 30. 2.13 Abbreviations, notations and nomenclature - p. 146 31. 3: Constitutive Models of Soft and Hard Living Tissues - p. 149 32. Abstract - p. 149 33. 3.1 Constitutive modeling - p. 149 34. 3.2 Isotropic elastic materials - p. 151 35. 3.3 Elasticity tensors - p. 154 36. 3.4 Isotropic hyperelastic materials - p. 156 37. 3.5 Incompressible materials - p. 159 38. 3.6 Compressible hyperelastic materials - p. 162 39. 3.7 Isotropic compressible hyperelasticity - p. 164 40. 3.8 Some forms of constitutive models - p. 165 41. 3.9 Saint-Venant Kirchhoff materials and Neo-Hookean materials through linearization - p. 167 42. 3.10 Hyperviscoelastic models - p. 170 43. 3.11 Anisotropic constitutive models: fiber reinforced solids, orthotropic materials - p. 174 44. 3.12 Case of orthotropic materials - p. 179 45. 3.13 Variational principles and hints to numerical solution schemes - p. 179 46. 4: Discrete Homogenization of Network Materials - p. 187 47. 5: Mechanics and Thermodynamics of Volumetric and Surface Growth - p. 211 48. Abstract - p. 211 49. 5.1 Introduction - p. 211 50. 5.2 Thermodynamics of continuous open media: a survey - p. 212 51. 5.3 General balance laws accounting for mass production due to growth - p. 216 52. 5.4 Growth kinematics and growth models - p. 224 53. 5.5 Mechanics and thermodynamics of surface growth - p. 225 54. 5.6 Surface growth: a review of surface thermodynamics - p. 236 55. 5.7 Material driving forces for surface growth - p. 249 56. 5.8 Extremum principles for biological continuum bodies undergoing volumetric and surface growth - p. 251 57. Part 2: Multiscale Bone Mechanics - p. 260 58. 6: Micropolar Models of Trabecular Bone - p. 263 59. Abstract - p. 263 60. 6.1 A survey of bone physiology - p. 263 61. 6.2 Review of trabecular bone models - p. 264 62. 6.3 Construction of two-dimensional anisotropic Cosserat bone models - p. 266 63. 6.4 Effective micropolar properties of bone - p. 269 64. 6.5 Numerical determination of the effective bone rigidities - p. 275 65. 6.6 Application to bone fracture - p. 283 66. 6.7 Effective mechanical properties of 3D vertebral trabecular bone - p. 287 67. 6.8 Results and discussion - p. 298 68. 6.9 Relevance of the choice of a micropolar effective continuum - p. 303 69. 6.10 Finite element validation - p. 306 70. 6.11 Discussion and limitations - p. 310 71. 7: Size-dependent Dynamic Behavior of Trabecular Bone - p. 317 72. 8: Prediction of Size Effects in Bone Brittle and Plastic Collapse - p. 345 73. 9: Multiscale Aspects of Bone Internal and External Remodeling - p. 389 74. Abstract - p. 389 75. 9.1 Introduction: biological mechanics of bone remodeling - p. 389 76. 9.2 Volumetric and surface growth model - p. 399 77. 9.3 Surface growth model - p. 408 78. 9.4 A coupled bone internal and external remodeling approach - p. 413 79. 9.5 Applications to hard tissues like bone - p. 421 80. 9.6 Macroscopic viewpoint - p. 430 81. 10: Integrated Remodeling to Fatigue Damage Model of Bone - p. 437 82. Part 3: Mechanics of Soft Biological Tissues: The Intervertebral Disk, Biological Networks, Ligaments and Tendons - p. 451 83. 11: Micromechanics of the Intervertebral Disk - p. 455 84. 12: Effective Mechanical Response of Biological Membranes - p. 469 85. 13: Micromechanics of Ligaments and Tendons - p. 489 86. Bibliography - p. 511 87. List of Authors - p. 555 88. Index - p. 557
Multiscale Biomechanics provides new insights on multiscale static and dynamic behavior of both soft and hard biological tissues, including bone, the intervertebral disk, biological membranes and tendons. The physiological aspects of bones and biological membranes are introduced, along with micromechanical models used to compute mechanical response. A modern account of continuum mechanics of growth and remodeling, generalized continuum models to capture internal lengths scales, and dedicated homogenization methods are provided to help the reader with the necessary theoretical foundations. Topics discussed include multiscale methods for fibrous media based on discrete homogenization, generalized continua constitutive models for bone, and a presentation of recent theoretical and numerical advances.
In addition, a refresher on continuum mechanics and more advanced background related to differential geometry, configurational mechanics, mechanics of growth, thermodynamics of open systems and homogenization methods is given in separate chapters. Numerical aspects are treated in detail, and simulations are presented to illustrate models.
This book is intended for graduate students and researchers in biomechanics interested in the latest research developments, as well as those who wish to gain insight into the field of biomechanics.
1. Introduction - p. xiii 2. Part 1: Theoretical Basis: Continuum Mechanics, Homogenization Methods, Thermodynamics of Growing Solid Bodies - p. xxi 3. 1: Tensor Calculus - p. 3 4. Abstract - p. 3 5. 1.1 A short historical vignette - p. 3 6. 1.2 Vector spaces - p. 5 7. 1.3 Covariant and contravariant tensors - p. 7 8. 1.4 Linear forms and duality - p. 8 9. 1.5 Tensor algebra - p. 11 10. 1.6 Euclidean tensors - p. 17 11. 1.7 Algebraic operations on tensors - p. 27 12. 1.8 Differential calculus on tensors: tensor analysis - p. 47 13. 1.9 Differential operators in curvilinear coordinates - p. 53 14. 1.10 Partial derivatives of function with tensor arguments - p. 65 15. 1.11 Elements of functional analysis - p. 71 16. 2: Continuum Mechanics - p. 75 17. Abstract - p. 75 18. 2.1 Motivations of nonlinear mechanics1 - p. 75 19. 2.2 Prerequisite: summary of linear elasticity - p. 83 20. 2.3 Introduction: notion of body in a continuum description - p. 85 21. 2.4 Kinematics: displacement, transformation gradient, strains - p. 86 22. 2.5 Deformation tensors - p. 100 23. 2.6 Polar decomposition theorem - p. 107 24. 2.7 Linearization of the kinematics: small strains and small displacements - p. 110 25. 2.8 Deformation velocities - p. 112 26. 2.9 Transport operations: pull-back and push-forward - p. 114 27. 2.10 Isotropic tensor functions - p. 118 28. 2.11 Stress measures and strain–strain duality - p. 121 29. 2.12 Balance laws - p. 137 30. 2.13 Abbreviations, notations and nomenclature - p. 146 31. 3: Constitutive Models of Soft and Hard Living Tissues - p. 149 32. Abstract - p. 149 33. 3.1 Constitutive modeling - p. 149 34. 3.2 Isotropic elastic materials - p. 151 35. 3.3 Elasticity tensors - p. 154 36. 3.4 Isotropic hyperelastic materials - p. 156 37. 3.5 Incompressible materials - p. 159 38. 3.6 Compressible hyperelastic materials - p. 162 39. 3.7 Isotropic compressible hyperelasticity - p. 164 40. 3.8 Some forms of constitutive models - p. 165 41. 3.9 Saint-Venant Kirchhoff materials and Neo-Hookean materials through linearization - p. 167 42. 3.10 Hyperviscoelastic models - p. 170 43. 3.11 Anisotropic constitutive models: fiber reinforced solids, orthotropic materials - p. 174 44. 3.12 Case of orthotropic materials - p. 179 45. 3.13 Variational principles and hints to numerical solution schemes - p. 179 46. 4: Discrete Homogenization of Network Materials - p. 187 47. 5: Mechanics and Thermodynamics of Volumetric and Surface Growth - p. 211 48. Abstract - p. 211 49. 5.1 Introduction - p. 211 50. 5.2 Thermodynamics of continuous open media: a survey - p. 212 51. 5.3 General balance laws accounting for mass production due to growth - p. 216 52. 5.4 Growth kinematics and growth models - p. 224 53. 5.5 Mechanics and thermodynamics of surface growth - p. 225 54. 5.6 Surface growth: a review of surface thermodynamics - p. 236 55. 5.7 Material driving forces for surface growth - p. 249 56. 5.8 Extremum principles for biological continuum bodies undergoing volumetric and surface growth - p. 251 57. Part 2: Multiscale Bone Mechanics - p. 260 58. 6: Micropolar Models of Trabecular Bone - p. 263 59. Abstract - p. 263 60. 6.1 A survey of bone physiology - p. 263 61. 6.2 Review of trabecular bone models - p. 264 62. 6.3 Construction of two-dimensional anisotropic Cosserat bone models - p. 266 63. 6.4 Effective micropolar properties of bone - p. 269 64. 6.5 Numerical determination of the effective bone rigidities - p. 275 65. 6.6 Application to bone fracture - p. 283 66. 6.7 Effective mechanical properties of 3D vertebral trabecular bone - p. 287 67. 6.8 Results and discussion - p. 298 68. 6.9 Relevance of the choice of a micropolar effective continuum - p. 303 69. 6.10 Finite element validation - p. 306 70. 6.11 Discussion and limitations - p. 310 71. 7: Size-dependent Dynamic Behavior of Trabecular Bone - p. 317 72. 8: Prediction of Size Effects in Bone Brittle and Plastic Collapse - p. 345 73. 9: Multiscale Aspects of Bone Internal and External Remodeling - p. 389 74. Abstract - p. 389 75. 9.1 Introduction: biological mechanics of bone remodeling - p. 389 76. 9.2 Volumetric and surface growth model - p. 399 77. 9.3 Surface growth model - p. 408 78. 9.4 A coupled bone internal and external remodeling approach - p. 413 79. 9.5 Applications to hard tissues like bone - p. 421 80. 9.6 Macroscopic viewpoint - p. 430 81. 10: Integrated Remodeling to Fatigue Damage Model of Bone - p. 437 82. Part 3: Mechanics of Soft Biological Tissues: The Intervertebral Disk, Biological Networks, Ligaments and Tendons - p. 451 83. 11: Micromechanics of the Intervertebral Disk - p. 455 84. 12: Effective Mechanical Response of Biological Membranes - p. 469 85. 13: Micromechanics of Ligaments and Tendons - p. 489 86. Bibliography - p. 511 87. List of Authors - p. 555 88. Index - p. 557
