Lecture 3Mechanics of Biomaterials

http://www.aeromech.usyd.edu.au/people/academic/qingli/MECH4981.htmCourse Web

• Establish biomaterial constitutive models

• Determine the biomechanical response to load

• Analyse the prosthetic design

• Estimate the health status of living tissues under stress

ObjectivesObjectives

Introductory Mechanics ModelIntroductory Mechanics Model

M

M

T

T

F

F

Recall “Lecture 1”: statics/dynamics methods to determine force/moment/torque

Introductory Mechanics Model – Stress AnalysisIntroductory Mechanics Model – Stress Analysis

z

y x

M M

Normal stress

?zz Motion Measurement

M

MT

TF

FDynamics analysis to

determine load

• Sport injury?• Bone damage?

Pure bending analysis

xx

xxzz I

yM

Methods of BiomechanicsMethods of Biomechanics

Analytical Method – Solid Mechanics I and II

Biomechanical Experiment – Test

Numerical Techniques – FEM

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 [] Strain tensor [] Stiffness tensor [S] (Stiffness tensor)

S

E

CS 1

Compliance tensor [C]=[S]-1

L L LL

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)

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

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

aFT

Uniaxial Test – Finite Large DeformationUniaxial Test – Finite Large Deformation

Density 0

1L

LLLl

TaF

L/l/m/m

aF

lL

VV

aF

lL

LAla

AF

Undef

def

11 0

0

AF

lLP

1

aFF

l Density

L L

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 – from tensor production

axial

lateral

StrainShearStressShear

G

VPV

V/VP

ePK

Sijijij 2

Relationship Between the Relationship Between the Elastic ConstantsElastic 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

21132

212

E

EGGEGG

122

21 EG

1232

GE

KG

1221123 GG

GGE

213

EK

Hooke’s Law – Tensor Representation Hooke’s Law – Tensor Representation

SC or:LawsHooke'

333231

232221

131211

OR:TensorStress

zzzyzx

yzyyyx

xzxyxx

333231

232221

131211

OR:TensorStrain

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 ijijklij S or

Hooke’s Law – Matrix Representation Hooke’s Law – Matrix Representation

C:LawsHooke'

12

13

23

33

22

11

12

13

23

33

22

11

12

13

23

33

22

11

12

13

23

33

22

11

666564636261

565554535251

464544434241

363534333231

262524212221

161514131211

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

CCompliance Matrix

Material Constitutive ModelsMaterial Constitutive Models

Anisotropy21 independent components elasticity matrix

Orthotropy9 independent components to elasticity matrix

Transverse isotropy5 independent components

Isotropy2 independent components

SC or:LawsHooke'

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

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

Symmetric matrix

Material Constitutive Models Material Constitutive Models –– Orthotropy Orthotropy

9 independent components to elasticity matrix (along 3 directions)

12

13

23

33

22

11

12

31

23

33

23

3

132

32

22

121

31

1

21

1

12

13

23

33

22

11

100000

010000

001000

0001

0001

0001

G

G

G

EEE

EEE

EEE

,,,G,G,G

,E,E,E

311332232112

312312

321

:RatiossPoisson'3:ModuliShear3

:ModulisYoung'3

1

2

3

Orthotropic Properties – Cortical Bone Orthotropic 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

ij: 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% : 30%

Material Constitutive Models – Transversely IsotropyMaterial Constitutive Models – Transversely Isotropy

5 independent components

1

23

12

1123231

12

3231

3

21

12

EGthatNoteGG

EEE

12

13

23

33

22

11

1

1231

31

33

31

3

311

31

11

121

31

1

12

1

12

13

23

33

22

11

1200000

010000

001000

0001

0001

0001

E

G

G

EEE

EEE

EEE

Material Constitutive Models – IsotropyMaterial Constitutive Models – Isotropy

2 independent components

12123231

123231

321

EGGGGthatNote

EEEE

1

2

3

12

13

23

33

22

11

12

13

23

33

22

11

1200000

0120000

0012000

0001

0001

0001

E

E

E

EEE

EEE

EEE

Hooke’s Law for an Isotropic Elastic MaterialHooke’s Law for an Isotropic Elastic Material

ijijijS 2:LawsHooke'

zxzxzxzx

yzyzyzyz

xyxyxyxy

zzzzyyxxzz

yyzzyyxxyy

xxzzyyxxxx

GG

GG

GG

G

G

G

22

22

22

2

2

2

zxzxzxzx

yzyzyzyz

xyxxxyxy

yyxxzzzz

xxzzyyyy

zzyyxxxx

GG

GG

GG

E

E

E

21

21

21

21

21

21

1

1

1Stress-Strain Relationship Strain-Stress Relationship

where ij – Kronecker delta, ij =1 if i=j, otherwise (i≠j), ij =0. That is

ijijijS 2:LawsHooke'

ijkkijij EE

1

zxzxzxzx

yzyzyzyz

xyxxxyxy

yyxxzzzz

xxzzyyyy

zzyyxxxx

GG

GG

GG

E

E

E

21

21

21

21

21

21

1

1

1

23233322112323

3322113322111111

21101

111

GEEE

EEE

e.g.

Hooke’s Law (Isotropic) – Cont’dHooke’s Law (Isotropic) – Cont’d

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

nzz

zt

tn

ntz

zn

t

tn

n

nt

nz

tz

zz

tt

nn

G

G

G

EEE

EEE

EEE

100000

010000

001000

0001

0001

0001

z (3)

x (1)

ez

et

en

AF

zz3

000

000

00

100000

010000

001000

0001

0001

0001

z

zz

z

zzzt

z

zzzn

zz

nt

zn

tz

zt

tz

n

nzz

zt

tn

ntz

zn

t

tn

n

nt

nz

tz

zz

tt

nn

E

E

E

G

G

G

EEE

EEE

EEE

F3 F3

Mechanics of Introductory Example – Cont’dMechanics of Introductory Example – Cont’d

Mechanics of Introductory Example – Cont’dMechanics of Introductory Example – Cont’d

ez

et Mxxz (3)

y (2) x (1)Myy

xx

xxzzxx I

yM:M toDue

yy

yyzzyy I

xM:M toDue

Pure Bending

Total stress in zz:

Eccentric Axial Loading

yy

yy

xx

xxzz I

xMI

yM

yyxxz

yy

yy

xx

xxzzz I

xx~

Iyy~

AF

IxM

IyM

AF 1

y~,x~

x

y

Equilibrium Equations (General)Equilibrium Equations (General)

0

0

0

3

2

1

bzyx

bzyx

bzyx

zzzyzx

yzyyyx

xzxyxx

TensorStress

zzzyzx

yzyyyx

xzxyxx

0 bdiv

Where:

div - Divergence

Dynamic equilibrium: ub div

Tb,b,b 321b

0 ij,ij b

Biomechanical Test MethodBiomechanical Test Method

Site-specific testFemoral neck test

Finite Element MethodFinite Element Method

Femur Knee Hip

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 modelFE modelFE model

PDLMolar

Part of modelComputationally more efficient

Whole Jaw modelComputationally more accurate

b) b) Sectional curvesSectional curves

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

FT

z

S

S

Section S-S

x

y

yh

l l

x

y

Cortical

CancellousR

rA

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%