mechanics of biomaterials
TRANSCRIPT
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%