Mechanics of Biomaterials
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This lecture discusses introductory mechanics models and constitutive models for biomaterials. The objectives are to establish biomaterial constitutive models, determine biomechanical response to load, analyze prosthetic design, and estimate health status of living tissues under stress. The lecture covers introductory mechanics modeling including stress analysis, normal and pure bending. Methods of biomechanics including analytical, experimental, and numerical techniques like FEM are presented. Constitutive models including elastic behavior, Hooke's law, elastic constants, material anisotropy, orthotropy, transverse isotropy, and isotropy are discussed.
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Mechanics of Biomaterials
- 1. Lecture 3 Mechanics of Biomaterials http://www.aeromech.usyd.edu.au/people/academic/qingli/MECH4981.htm Course Web
- 2. • Establish biomaterial constitutive models • Determine the biomechanical response to load • Analyse the prosthetic design • Estimate the health status of living tissues under stress ObjectivesObjectives
- 3. Introductory Mechanics ModelIntroductory Mechanics Model M M T T F F Recall “Lecture 1”: statics/dynamics methods to determine force/moment/torque
- 4. Introductory Mechanics Model – Stress AnalysisIntroductory Mechanics Model – Stress Analysis z y x M M Normal stress [ ]?zz σσ ≤ Motion Measurement M M T T F F Dynamics analysis to determine load • Sport injury? • Bone damage? Pure bending analysis xx xx zz I yM =σ
- 5. Methods of BiomechanicsMethods of Biomechanics Analytical Method – Solid Mechanics I and II Biomechanical Experiment – Test Numerical Techniques – FEM
- 6. Elastic BehaviorElastic Behavior Basic element representing an elastic material Hooke’s law, Young’s modulus, Poisson’s ratio etc Hooke’s Law (uniaxial): • the strain is directly proportional to the stress Hooke’s Law (General): • Stress tensor [σ] • Strain tensor [ε] • Stiffness tensor [S] (Stiffness tensor) [ ] [ ] [ ]εσ S= εσ E= [ ] [ ] [ ] [ ][ ]σσε CS == −1 • Compliance tensor [C]=[S]-1 ∆L L L L∆ ε =
- 7. Elastic Constants – Young’s ModulusElastic Constants – Young’s Modulus Young’s Modulus E: • Relationship between tensile or compressive stress and strain • Applies for small strains (within the elastic range) εσ E= * http://www.lib.umich.edu/dentlib/Dental_tables/toc.html Biomaterials (Isotropic) E (GPa)* Cancellous bone 0.49 Cortical bone 14.7 Long bone - Femur 17.2 Long bone - Humerus 17.2 Long bone - Radius 18.6 Long bone - Tibia 18.1 Vertebrae - Cervical 0.23 Vertebrae - Lumbar 0.16
- 8. Undeformed Configuration • length = L • Undeformed area = A Deformed Configuration • length = l • Deformed area = a Cauchy Stress (True stress) Nominal Stress (Engineering Stress) Second Piola-Kirchhoff Stress L A a F T = Uniaxial Test – Finite Large DeformationUniaxial Test – Finite Large Deformation Density ρ 0 ε ∆ ϑ += + == 1 L LL L l T a F L/l/m /m a F l L V V a F l L LA la A F Undef def ϑρ ρ ρ ρ σ 11 0 0 ===== A F l L P ϑ σ 1 == a FF l Density ρ L ∆L
- 9. Elastic Constants –Elastic Constants – (other 4 constants)(other 4 constants) Poisson’s ratio Describe lateral deformation in response to an axial load Shear Modulus Describes relationship between applied torque and angle of deformation Bulk Modulus Describes the change in volume in response to hydrostatic pressure (equal stresses in all directions) Lame’s constant λ – from tensor production axial lateral ε ε ν −= StrainShear StressShear G == γ τ V P V V/V P e P K ∂ ∂ −≈−=−= ∆ ∆∆ [ ] [ ] [ ]εσ S= ijijij µεδλεσ αα 2+=
- 10. Relationship Between theRelationship Between the Elastic ConstantsElastic Constants Young’s modulus (E) Poisson’s ratio (ν) Bulk modulus (K) Shear modulus (G) Lame’s constant (λ) For an isotropic material, elastic constants are CONSTANT ( ) ( )( )νν ν ν ν λ 2113 2 21 2 −+ = − − = − = E EG GEGG ( ) ( )νν νλ + = − = 122 21 E G ( ) ( ) 1 232 −= − = + = G E KG λ λ λ λ ν ( ) ( )( ) ( )ν ν ννλ λ λ += −+ = + + = 12 21123 G G GG E ( )ν213 − = E K
- 11. Hooke’s Law – Tensor RepresentationHooke’s Law – Tensor Representation [ ] [ ][ ] [ ] [ ][ ]εσσε SC == or:LawsHooke' [ ] [ ] = = 333231 232221 131211 OR:TensorStress σσσ σσσ σσσ σ σσσ σσσ σσσ σ zzzyzx yzyyyx xzxyxx [ ] [ ] = = 333231 232221 131211 OR:TensorStrain εεε εεε εεε ε εεε εεε εεε ε zzzyzx yzyyyx xzxyxx Remarks: • Stress tensor and strain tensor are the 2nd order tensors • [S] and [C] are the fourth order tensor (1 x, 2 y, 3 z) ijijklij C σε = ijijklij S εσ =or
- 12. Hooke’s Law – Matrix RepresentationHooke’s Law – Matrix Representation { } [ ]{ }σε C=:LawsHooke' { } = = 12 13 23 33 22 11 12 13 23 33 22 11 τ τ τ σ σ σ σ σ σ σ σ σ σ{ } = = 12 13 23 33 22 11 12 13 23 33 22 11 γ γ γ ε ε ε ε ε ε ε ε ε ε [ ] = 666564636261 565554535251 464544434241 363534333231 262524212221 161514131211 CCCCCC CCCCCC CCCCCC CCCCCC CCCCCC CCCCCC C Compliance Matrix
- 13. Material Constitutive ModelsMaterial Constitutive Models Anisotropy 21 independent components elasticity matrix Orthotropy 9 independent components to elasticity matrix Transverse isotropy 5 independent components Isotropy 2 independent components [ ] [ ][ ] [ ] [ ][ ]εσσε SC == or:LawsHooke'
- 14. Material Constitutive Models – AnisotropyMaterial Constitutive Models – Anisotropy (Most likely) 21 independent components in elasticity matrix = 12 13 23 33 22 11 665646362616 565545352515 464544342414 363534332313 262524212212 161514131211 12 13 23 33 22 11 σ σ σ σ σ σ ε ε ε ε ε ε CCCCCC CCCCCC CCCCCC CCCCCC CCCCCC CCCCCC Symmetric matrix
- 15. Material Constitutive ModelsMaterial Constitutive Models –– OrthotropyOrthotropy 9 independent components to elasticity matrix (along 3 directions) −− −− −− = 12 13 23 33 22 11 12 31 23 33 23 3 13 2 32 22 12 1 31 1 21 1 12 13 23 33 22 11 1 00000 0 1 0000 00 1 000 000 1 000 1 000 1 σ σ σ σ σ σ νν νν νν ε ε ε ε ε ε G G G EEE EEE EEE ,,, G,G,G ,E,E,E 311332232112 312312 321 :RatiossPoisson'3 :ModuliShear3 :ModulisYoung'3 νννννν === 1 2 3
- 16. Orthotropic Properties – Cortical BoneOrthotropic Properties – Cortical Bone E1: 6.91 - 18.1 GPa E2 : 8.51 - 19.4 GPa E3 : 17.0 - 26.5 GPa G12: 2.41 - 7.22 GPa G13: 3.28 - 8.65 GPa G23: 3.28 - 8.67 GPa νij: 0.12 - 0.62 Young’s Moduli Shear Moduli Poisson’s Ratios Remarks: the high standard deviations in property values seen in one are not necessarily (although may possibly be) due to experimental error • E: 15% • G: 10% • ν : 30%
- 17. Material Constitutive Models – Transversely IsotropyMaterial Constitutive Models – Transversely Isotropy 5 independent components 1 2 3 ( )12 1 123231 12 3231 3 21 12 ν ν νν + ==• • =• • =• E GthatNoteGG E EE ( ) + −− −− −− = 12 13 23 33 22 11 1 12 31 31 33 31 3 31 1 31 11 12 1 31 1 12 1 12 13 23 33 22 11 12 00000 0 1 0000 00 1 000 000 1 000 1 000 1 σ σ σ σ σ σ ν νν νν νν ε ε ε ε ε ε E G G EEE EEE EEE
- 18. Material Constitutive Models – IsotropyMaterial Constitutive Models – Isotropy 2 independent components ( )ν νννν + ==== ===• ===• 12 123231 123231 321 E GGGGthatNote EEEE 1 2 3 ( ) ( ) ( ) + + + −− −− −− = 12 13 23 33 22 11 12 13 23 33 22 11 12 00000 0 12 0000 00 12 000 000 1 000 1 000 1 σ σ σ σ σ σ ν ν ν νν νν νν ε ε ε ε ε ε E E E EEE EEE EEE
- 19. Hooke’s Law for an Isotropic Elastic MaterialHooke’s Law for an Isotropic Elastic Material [ ] [ ][ ] ijijijS µεδλεσεσ αα 2:LawsHooke' +=⇔= ( ) ( ) ( ) == == == +++= +++= +++= zxzxzxzx yzyzyzyz xyxyxyxy zzzzyyxxzz yyzzyyxxyy xxzzyyxxxx GG GG GG G G G 22 22 22 2 2 2 τεσ τεσ γτεσ εεεελσ εεεελσ εεεελσ ( )[ ] ( )[ ] ( )[ ] == == == +−= +−= +−= zxzxzxzx yzyzyzyz xyxxxyxy yyxxzzzz xxzzyyyy zzyyxxxx GG GG GG E E E τγσε τγσε τγσε σσνσε σσνσε σσνσε 2 1 2 1 2 1 2 1 2 1 2 1 1 1 1 Stress-Strain Relationship Strain-Stress Relationship
- 20. where δij – Kronecker delta, δij =1 if i=j, otherwise (i≠j), δij =0. That is [ ] [ ][ ] ijijijS µεδλεσεσ αα 2:LawsHooke' +=⇔= ijkkijij EE δσ ν σ ν ε − + = 1 ( )[ ] ( )[ ] ( )[ ] == == == +−= +−= +−= zxzxzxzx yzyzyzyz xyxxxyxy yyxxzzzz xxzzyyyy zzyyxxxx GG GG GG E E E τγσε τγσε τγσε σσνσε σσνσε σσνσε 2 1 2 1 2 1 2 1 2 1 2 1 1 1 1 ( ) ( )[ ] ( ) 23233322112323 3322113322111111 2 11 0 1 1 1 1 σσ ν σσσ ν σ ν ε σσνσσσσ ν σ ν ε GEEE EEE = + =×++− + = +−=×++− + = e.g. Hooke’s Law (Isotropic) – Cont’dHooke’s Law (Isotropic) – Cont’d
- 21. Mechanics Model of Introductory ExampleMechanics Model of Introductory Example z (3) y (2) x (1) ez et en −− −− −− = nt nz tz zz tt nn nt zn tz zt tz n nz z zt tn nt z zn t tn n nt nz tz zz tt nn G G G EEE EEE EEE σ σ σ σ σ σ νν νν νν ε ε ε ε ε ε 1 00000 0 1 0000 00 1 000 000 1 000 1 000 1
- 22. z (3) x (1) ez et en A F zz 3 −=σ − − = −− −− −− = 0 0 0 0 0 0 0 0 1 00000 0 1 0000 00 1 000 000 1 000 1 000 1 z zz z zzzt z zzzn zz nt zn tz zt tz n nz z zt tn nt z zn t tn n nt nz tz zz tt nn E E E G G G EEE EEE EEE σ σν σν σ νν νν νν ε ε ε ε ε ε F3 F3 Mechanics of Introductory Example – Cont’dMechanics of Introductory Example – Cont’d
- 23. Mechanics of Introductory Example – Cont’dMechanics of Introductory Example – Cont’d ez et Mxx z (3) y (2) x (1) Myy xx xx zzxx I yM :M =σtoDue yy yy zzyy I xM :M =σtoDue Pure Bending Total stress in zz: Eccentric Axial Loading yy yy xx xx zz I xM I yM ±=σ ±+−=±+−= yyxx z yy yy xx xxz zz I xx~ I yy~ A F I xM I yM A F 1 σ ( )y~,x~ x y
- 24. Equilibrium Equations (General)Equilibrium Equations (General) 0 0 0 3 2 1 =+ ∂ ∂ + ∂ ∂ + ∂ ∂ =+ ∂ ∂ + ∂ ∂ + ∂ ∂ =+ ∂ ∂ + ∂ ∂ + ∂ ∂ b zyx b zyx b zyx zzzyzx yzyyyx xzxyxx σσσ σσσ σσσ [ ] TensorStress = zzzyzx yzyyyx xzxyxx σσσ σσσ σσσ σ [ ] 0=+ bσdiv Where: div - Divergence Dynamic equilibrium: [ ] ub ρ=+σdiv [ ]T b,b,b 321=b 0=+ ij,ij bσ
- 25. Biomechanical Test MethodBiomechanical Test Method Site-specific testFemoral neck test
- 26. Finite Element MethodFinite Element Method Femur Knee Hip
- 27. CT-Based Finite Element Modelling ProcedureCT-Based Finite Element Modelling Procedure a) CT Image Segmentationa) CT Image Segmentation c) CAD modelc) CAD model d) FE modeld) FE model FE modelFE model PDL Molar Part of model Computationally more efficient Whole Jaw model Computationally more accurate b)b) Sectional curvesSectional curves
- 28. Finite Element Modelling ExampleFinite Element Modelling Example 3 unit all-ceramic dental bridge analysis3 unit all-ceramic dental bridge analysis Solid model VM stress Contour
- 29. FT z S S Section S-S x y yh l l x y Cortical Cancellous R r A B AssignmentAssignment Approximately use engineering beam theory to calculate principal stresses – 60% • Mohr circles • Nature of stress (tension or compression) Apply 3D finite element method to calculate the principal stress – 30% • Selection of elements and mesh density • Contours of principal stress • Comparison against analytical solution from Beam Theory Fixed M Submission of tutorial question of callus formation mechanics – 10%