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NSF-JHU Workshop on Probability & Materials Multi-Scale Micro-Mechanical Poroelastic Modeling of Fluid Flow in Cortical Bone Colby C. Swan, HyungJoo Kim Center for Computer-Aided Design The University of Iowa Iowa City, Iowa USA WORKSHOP ON PROBABILITY AND MATERIALS: FROM NANO- TO MACRO-SCALE Johns Hopkins University Baltimore, MD, USA January 5-7, 2005
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NSF-JHU Workshop on Probability & Materials BACKGROUND Bones are adaptive living tissues: Size & density increase and decrease with the intensity of mechanical loading Intensified loading within certain frequency range (10-100 Hz) → bone growth Static loading does not stimulate bone growth Reduced loading → bone atrophy
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NSF-JHU Workshop on Probability & Materials BACKGROUND (CONT’D) Understanding the precise mechanisms in bone adaptation is an important objective in orthopedic research. Potentially significant clinical implications Provide necessary stimulus to maintain bone mass Loss of bone mass is impediment to human travel in space
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NSF-JHU Workshop on Probability & Materials Osteocyte process Fibrous glycocalyx matrix Unit cell of lacuna and canaliculi 50-100 m 30 m 0.1 m ~ 1 cm Hierarchical Structure of Whole Bone
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NSF-JHU Workshop on Probability & Materials Transverse section of cortical bone
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NSF-JHU Workshop on Probability & Materials Transverse section of cortical bone
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NSF-JHU Workshop on Probability & Materials Transverse section of cortical bone
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NSF-JHU Workshop on Probability & Materials GOALS OF CURRENT RESEARCH Quantify nature of load-induced fluid flow in lacunar canalicular system; Magnitude of fluid pressures & shear stresses Compare pressures & stresses with those known to stimulate osteocyte response in in vitro cell culture studies. Adequately capture the heterogeneity and hierarchy of cortical bone. Address relevant multi-scale modeling challenges
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NSF-JHU Workshop on Probability & Materials Multi-Scale Modeling X L denotes coordinate scale of lacunae ~ 50 m X C denotes coordinate scale of canaliculus ~ 0.5 m
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NSF-JHU Workshop on Probability & Materials Osteocyte process Fibrous glycocalyx matrix MICROMECHANICS & HOMOGENIZATION OF CANALICULAR BONE Temporary idealization solid matrix fluid-filled canal
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NSF-JHU Workshop on Probability & Materials SCALE BRIDGING: CANALICULAR TO LACUNAR SCALE KINEMATICS: Total strain rate Rate of fluid content change
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NSF-JHU Workshop on Probability & Materials SCALE BRIDGING: CANALICULAR TO LACUNAR SCALE STRESSES: “Solid” stress tensor “Fluid” stress tensor “Total” stress tensor
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NSF-JHU Workshop on Probability & Materials Linear Poroelasticity Model (After Biot 1941, 1962) C is the 6x6 symmetric undrained elastic matrix G is the 6x1 strain-pore-pressure coupling matrix Z is the scalar storage modulus Procedure for computing C, G, Z: Impose and on model; Compute and ;
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NSF-JHU Workshop on Probability & Materials Effective poroelastic properties of canalicular bone matrix X 3 direction is taken as aligned with the canaliculus. ; D canaliculi = 0.10 m Storage Modulus (Gpa) Pore-pressure Coupling Moduli (Gpa) Undrained Moduli (Gpa)
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NSF-JHU Workshop on Probability & Materials <<1 where Osteocytic process Bone matrix Fibrous glycocalyx matrix Hydraulic Conductivity of Canalicular System Assumption 1: Canaliculus fully occupied by fluid Assumption 2: Canaliculus occupied by osteocytic process and glycocalyx matrix. (Tsay and Weinbaum, 1991) (Weinbaum et al, 1994)
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NSF-JHU Workshop on Probability & Materials Axial permeability (m 2 ) Transverse permeability (m 2 ) Clear canaliculus assumption 6.7·10 -18 m 2 6.7·10 -21 m 2 Glycocalyx matrix assumption 7.5·10 -20 m 2 6.7·10 -21 m 2 HYDRAULIC CONDUCTIVITIES USED IN COMPUTATIONS
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NSF-JHU Workshop on Probability & Materials Poroelastic Modeling at Lacunar Scale Total Medium Momentum Balance Fluid Equation of Motion Strained-Controlled Poroelastic Unit Cell Analysis Method: Impose history of and on unit cell model; Compute corresponding histories of.
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NSF-JHU Workshop on Probability & Materials Modeling of Fluid Lacunar Region (Stokes Flow Model) Interface Conditions: (Between Lacuna and Poroelastic Region)
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NSF-JHU Workshop on Probability & Materials Computed fluid pressure distributions under undrained loading of lacunar unit cell model (Stresses are in 10 6 dynes/cm 2 )
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NSF-JHU Workshop on Probability & Materials Fluid Pressure Distribution Under Harmonic Shear Loading
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NSF-JHU Workshop on Probability & Materials Fluid Pressure Distributions Harmonic Extensional Loading
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NSF-JHU Workshop on Probability & Materials Shear loadingAxial loading Local Fluid Flows in Vicinity of Lacuna:
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NSF-JHU Workshop on Probability & Materials Energy Considerations Stored and Dissipated Energies Per Unit Volume
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NSF-JHU Workshop on Probability & Materials
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Modeling of Multiple Lacunae at Osteonal Scale
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NSF-JHU Workshop on Probability & Materials OSTEONAL MODEL 4% Haversian porosity 4% Lacunar porosity 2% Canalicular porosity FEM Mesh of Model Canalicular orientations in bone matrix
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NSF-JHU Workshop on Probability & Materials Fluid Pressure Distribution Under Harmonic Shear Loading of Osteonal Unit Cell Model 12 =10 -4 ; f = 1 Hz;
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NSF-JHU Workshop on Probability & Materials
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Summary & Conclusions Significant fluid pressures and flow do occur in the lacunar-canalicular system: At physiologically meaningful frequencies ( 10 0 Hz < f < 10 2 Hz); At physiologically meaningful strains ~ O(10 -5 ); Remaining challenges (among many): Begin to incorporate osteocyte (cell-mechanics) models; Incorporate chemical transport phenomena;
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NSF-JHU Workshop on Probability & Materials Multi-Scale Analysis Process Plain-Weave Textile Composite Hex-arranged fiber Homogenization and Material Modeling of Yarn Composite Aligned Fiber Composite Structure Homogenization and Material Modeling of Textile Composite
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NSF-JHU Workshop on Probability & Materials Material Scale Decompositions of Stress and Deformation Decomposition material/ macro stress/ deformation Periodicity of material scale stress and deformation fields:
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NSF-JHU Workshop on Probability & Materials Computational Unit-Cell Homogenization Deformation controlled unit-cell analysis - Apply macro-scale deformation field solve for periodic displacement field ; such that satisfies stress equilibrium. Homogenized macro-scale stress / deformation Determine homogenized material constitutive model out of macroscopic stress and deformation relations;
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NSF-JHU Workshop on Probability & Materials Computational Unit-Cell Homogenization Deformation controlled unit-cell analysis - Apply macro-scale deformation field solve for periodic displacement field ; such that satisfies stress equilibrium. Homogenized macro-scale stress / deformation Determine homogenized material constitutive model out of macroscopic stress and deformation relations;
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NSF-JHU Workshop on Probability & Materials Symmetric and Conjugate Stress, Strain Measures Conjugacy between averaged deformation gradient and averaged nominal stress was demonstrated by Nemat-Nasser (2000). Modified symmetric stress and strain measures satisfy conjugacy in strain energy Macro 2 nd P-K Stress Tensor; Macro Green-Lagrange Strain Tensor
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NSF-JHU Workshop on Probability & Materials Transversely-Isotropic Hyperelastic Model Transversely-Isotropic Hyper-elastic model by Bonet and Burton (1998) where and A is material director of aligned fibers in material configuration A: material director yarn
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NSF-JHU Workshop on Probability & Materials Estimation of Coefficients for Aligned Fiber Model Coefficients for Bonet and Burton model Deformation-controlled homogenization where is homogenized stress under k th strain-controlled case is 2 nd PK stress from proposed model.
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NSF-JHU Workshop on Probability & Materials Unit-Cell of Hexagonally Packed Aligned Fibers (a) In-plane compression (b) In-plane tension(c) Out-of-plane shear (d) In-plane shear undeformed x1x1 x2x2 x3x3
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NSF-JHU Workshop on Probability & Materials Stress-Strain Behavior of Aligned-Fiber Composite E 11 S (GPa) S 11 S 11 Fit S 33 (S 33 Fit) S 22 (S 22 Fit) 11 ≠ 0, others zero
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NSF-JHU Workshop on Probability & Materials Unit Cell of Plain-Weave Textile Composite (a) Unit Cell for Plain-Weave Textile Composite (b) Quarter of Unit Cell Periodic Displacement Condition Traction Free x1x1 x2x2 x3x3
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NSF-JHU Workshop on Probability & Materials Deformed Shapes of Plain-weave Textile (a) uni-axial compression Loading x1x1 x2x2 x3x3 (b) uni-axial stretch x1x1 x2x2 x3x3 Loading
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