Mechanics of Biomaterials Course Web http://www.aeromech.usyd.edu.au/people/academic/qingli/MECH4981.htm

Objectives Establish biomaterial constitutive models Determine the biomechanical response to load Analyse the prosthetic design Estimate the health status of living tissues under stress

Introductory Mechanics Model F M T Recall “Lecture 1”: statics/dynamics methods to determine force/moment/torque T M F

Dynamics analysis to determine load Introductory Mechanics Model – Stress Analysis Sport injury? Bone damage? Normal stress Motion Measurement Pure bending analysis M T F M y x z Dynamics analysis to determine load

Methods of Biomechanics Analytical Method – Solid Mechanics I and II Biomechanical Experiment – Test Numerical Techniques – FEM

Elastic 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 []  Strain tensor []  Stiffness tensor [S] (Stiffness tensor) L L  Compliance tensor [C]=[S]-1

Elastic Constants – Young’s Modulus Young’s Modulus E: Relationship between tensile or compressive stress and strain Applies for small strains (within the elastic range) Biomaterials (Isotropic) E (GPa)* Cancellous bone 0.49 Cortical bone 14.7 Long bone - Femur 17.2 Long bone - Humerus Long bone - Radius 18.6 Long bone - Tibia 18.1 Vertebrae - Cervical 0.23 Vertebrae - Lumbar 0.16 * http://www.lib.umich.edu/dentlib/Dental_tables/toc.html

Uniaxial Test – Finite Large Deformation Undeformed Configuration  length = L  Undeformed area = A Deformed Configuration  length = l  Deformed area = a L A Density  0 a F l Density  L L Cauchy Stress (True stress) Nominal Stress (Engineering Stress) Second Piola-Kirchhoff Stress

Elastic 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  – from tensor production

Relationship Between the Elastic Constants Young’s modulus (E) Poisson’s ratio () Bulk modulus (K) Shear modulus (G) Lame’s constant () For an isotropic material, elastic constants are CONSTANT

Hooke’s Law – Tensor Representation (1  x, 2  y, 3  z) or Remarks: Stress tensor and strain tensor are the 2nd order tensors [S] and [C] are the fourth order tensor

Hooke’s Law – Matrix Representation Compliance Matrix

Material Constitutive Models Anisotropy 21 independent components elasticity matrix Orthotropy 9 independent components to elasticity matrix Transverse isotropy 5 independent components Isotropy 2 independent components

Material Constitutive Models – Anisotropy (Most likely) 21 independent components in elasticity matrix Symmetric matrix

Material Constitutive Models – Orthotropy 9 independent components to elasticity matrix (along 3 directions) 1 2 3

Orthotropic Properties – Cortical Bone Young’s Moduli 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 ij: 0.12 - 0.62 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%   : 30%

Material Constitutive Models – Transversely Isotropy 5 independent components 1 2 3

Material Constitutive Models – Isotropy 2 independent components 1 2 3

Hooke’s Law for an Isotropic Elastic Material Stress-Strain Relationship Strain-Stress Relationship

Hooke’s Law (Isotropic) – Cont’d where ij – Kronecker delta, ij =1 if i=j, otherwise (i≠j), ij =0. That is e.g.

Mechanics Model of Introductory Example en x (1) ez z (3) et

Mechanics of Introductory Example – Cont’d en x (1) ez F3 F3 z (3) et

Mechanics of Introductory Example – Cont’d Pure Bending y (2) x (1) Myy ez z (3) et Mxx Total stress in zz: x y Eccentric Axial Loading

Equilibrium Equations (General) Where: div - Divergence Dynamic equilibrium:

Biomechanical Test Method Femoral neck test Site-specific test

Finite Element Method Femur Knee Hip

CT-Based Finite Element Modelling Procedure Molar PDL FE model a) CT Image Segmentation b) Sectional curves c) CAD model d) FE model Whole Jaw model Computationally more accurate Part of model Computationally more efficient

3 unit all-ceramic dental bridge analysis Finite Element Modelling Example 3 unit all-ceramic dental bridge analysis Solid model VM stress Contour

Assignment 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 y Section S-S y T F Fixed S B Cancellous yh R z x A r S M x Cortical l l Submission of tutorial question of callus formation mechanics – 10%