# Mechanics of Biomaterials

Mechanics

Jul 18, 2012 (5 years and 10 months ago)

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Lecture 2
Mechanics of Biomaterials
Course Web
•Establish biomaterial constitutive models
•Determine the biomechanical response to load
•Analysethe prosthetic design
•Estimate the health status of living tissues under stress
Objectives
Objectives
Introductory Mechanics Model
Introductory Mechanics Model
M
M
T
T
F
Recall “Lecture 1”:
statics/dynamics
methods to determine
force/moment/torque
F
Introductory Mechanics Model
Introductory Mechanics Model

Stress Analysis
Stress Analysis
z
y
x
M
MNormal stress
[
]
?
zz
σ
σ

Motion Measurement
M
M
T
T
F
F
Dynamics analysis to
•Sport injury?
•Bone damage?
Pure bending analysis
xx
xx
zz
I
yM
=
σ
Methods of Biomechanics
Methods of Biomechanics
Analytical Method –
Solid Mechanics I and II
Biomechanical Experiment –
Test
Numerical Techniques –FEM
Elastic Behavior
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 [σ]
•Strain tensor [ε]
•Stiffness tensor [S] (Stiffness tensor)
[
][]
[
]
ε
σ
S
=
ε
σ
E
=
[][]
[
]
[
]
[
]
σσε
CS==
−1
•Compliance tensor [C]=[S]-1
Elastic Constants
Elastic Constants

Young
Young

s Modulus
s Modulus
Young’s ModulusE:
•Relationship between tensile orcompressive stress and strain
•Applies for small strains (within the elasticrange)
ε
σ
E
=
* http://www.lib.umich.edu/dentlib/Dental_tables/toc.html
Biomaterials (Isotropic)E (GPa)*
Cancellousbone0.49
Cortical bone14.7
Long bone -Femur17.2
Long bone -Humerus17.2
Long bone -Tibia18.1
Vertebrae -Cervical0.23
Vertebrae -Lumbar0.16
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

−≈−=−=
Δ
Δ
Δ
[
]
[
]
[
]
ε
σ
S
=
ijijij
μ
ε
δ
λ
ε
σ
αα
2
+
=
Relationship Between the
Relationship Between the
Elastic Constants
Elastic Constants
Young’s modulus (E)
Poisson’s ratio(ν)
Bulk modulus(K)
Shear modulus(G)
Lame’s constant(λ)
For an isotropic material, elasticconstants are CONSTANT
()
()()
νν
ν
ν
ν
λ
2113
2
21
2
−+
=

=

=
E
EG
GEGG
(
)
()
νν
ν
λ
+
=

=
122
21E
G
()
()
1
232
−=

=
+
=
G
E
KG
λ
λ
λ
λ
ν
(
)
(
)
(
)
()
ν
ν
ν
ν
λ
λ
λ
+=

+
=
+
+
=12
21123
G
G
GG
E
()
ν
213−
=
E
K
Hooke
Hooke

s Law
s Law

Tensor Representation
Tensor Representation
[
]
[
]
[
]
[
]
[
][]
ε
σ
σ
ε
SC
=
=
or:LawsHooke'
[][]

=

=
333231232221
131211
OR:TensorStress
σσσσσσ
σσσ
σ
σσσ
σσσσσσ
σ
zzzyzx
yzyyyx
xzxyxx
[][]

=

=
333231232221
131211
OR:TensorStrain
εεεεεεεεε
ε
εεε
εεεεεε
ε
zzzyzxyzyyyx
xzxyxx
Remarks:
•Stress tensor and strain tensor are the 2
nd
order tensors

[S]
and [C]
are the fourth order tensor
(1 ⎯x, 2 ⎯y, 3 ⎯z)
ijijklij
C
σ
ε
=
ijijklij
S
ε
σ
=
or
Hooke
Hooke

s Law
s Law

Matrix Representation
Matrix Representation
{
}
[
]
{
}
σ
ε
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
[]

=
666564636261565554535251464544434241363534333231262524212221
161514131211
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
C
Material Constitutive Models
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
[
]
[
]
[
]
[
]
[
]
[
]
ε
σ
σ
ε
SC
=
=
or:LawsHooke'
Material Constitutive Models
Material Constitutive Models

Anisotropy
Anisotropy
(Most likely) 21 independent components in elasticity matrix

=

12
13
23
33
22
11
665646362616565545352515464544342414363534332313262524212212
161514131211
12
13
23
33
22
11
σ
σ
σ
σ
σ
σ
ε
ε
ε
ε
ε
ε
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
CCCCCC
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
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
ν
ν
ν
ν
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
Orthotropic Properties
Orthotropic Properties

Cortical Bone
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
ν
ij: 0.12 -0.62
Remarks: the high standard deviations in property values seen inone are
not necessarily (although may possibly be) due to experimental error
•E: 15%
•G: 10%
•ν: 30%
Young’s Moduli
Shear Moduli
Poisson’s Ratios
Material Constitutive Models
Material Constitutive Models

Transversely Isotropy
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
ν
ν
ν
ν
ε
ε
ε
ε
ε
ε
E
G
G
EEE
EEE
EEE
Material Constitutive Models
Material Constitutive Models

Isotropy
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
Hooke
Hooke

s Law for an Isotropic Elastic Material
s Law for an Isotropic Elastic Material
[
]
[
]
[
]
ijijij
S
μ
ε
δ
λ
ε
σ
ε
σ
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
Strain-Stress Relationship
Stress-Strain Relationship
where
δ
ij
–Kroneckerdelta,
δ
ij
=1 ifi=j, otherwise (i≠j),
δ
ij
=0. That is
[
]
[
]
[
]
ijijij
S
μ
ε
δ
λ
ε
σ
ε
σ
αα
2:LawsHooke'
+
=

=
ijkkijij
EE
δσ
ν
σ
ν
ε

+
=
1
()
[]
()
[]
()
[]

==
==
==
+−=
+−=
+−=
zxzxzxzx
yzyzyzyz
xyxxxyxy
yyxxzzzz
xxzzyyyy
zzyyxxxx
GG
GG
GG
E
E
E
τγσε
τγσετγσε
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
Hooke

s Law (Isotropic)
s Law (Isotropic)

Cont
Cont

d
d
Mechanics Model of Introductory Example
Mechanics 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
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
t
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
Mechanics of Introductory Example

Cont
Cont

d
d
Mechanics of Introductory Example
Mechanics of Introductory Example

Cont
Cont

d
d
ez
et
Mxx
z (3)
y (2)
x (1)
Myy
xx
xx
zzxx
I
yM
:M=
σ
toDue
yy
yy
zzyy
I
xM
:M=
σ
toDue
Pure Bending
Total stress in zz:
yy
yy
xx
xx
zz
I
xM
I
yM
±=
σ

±+−=±+−=
yyxx
z
yy
yy
xx
xx
z
zz
I
xx
~
I
yy
~
A
F
I
xM
I
yM
A
F
1
σ
(
)
y
~
,x
~
x
y
Equilibrium Equations (General)
Equilibrium Equations (General)
0
0
0
3
2
1
=+

+

+

=+

+

+

=+

+

+

b
zyx
b
zyx
b
zyx
zz
zy
zx
yzyyyx
xz
xy
xx
σ
σ
σ
σσσ
σ
σ
σ
[]
TensorStress

=
zzzyzx
yzyyyxxzxyxx
σσσ
σσσ
σσσ
σ
[
]
0
=
+
b
σ
div
Where:
div-Divergence
Dynamic equilibrium:
[
]
ub
&&
ρ
=
+
σ
div
[
]
T
b,b,b
321
=b
0
=
+
ij,ij
b
σ
Biomechanical Test Method
Biomechanical Test Method
Site-specific testFemoral neck test
Finite Element Method
Finite Element Method
Femur Knee
Hip
CT
CT
-
-
Based Finite Element
Based Finite Element
Modelling
Modelling
Procedure
Procedure
a) CT Image Segmentation
a) CT Image Segmentation
d) FE model
d) FE model
FE model
FE model
PDL
Molar
Part of model
Computationally more efficient
Whole Jaw model
Computationally more accurate
b)
b)
Sectional curves
Sectional curves
Finite Element
Finite Element
Modelling
Modelling
Example
Example
3 unit all
3 unit all
-
-
ceramic dental bridge analysis
ceramic dental bridge analysis
Solid model
VM stress Contour
F
T
z
S
S
Section S-S
x
y
yh
l
l
x
y
Cortical
Cancellous
R
r
A
B
Assignment
Assignment
Approximately use engineering beam theory to calculate principal stresses –80%

Mohr circles

Nature of stress (tension or compression)
Apply 3D finite element method to calculate the principal stress –20%

Selection of elements and mesh density

Contours of principal stress

Comparison against analytical solution from Beam Theory
Fixed
M
30°