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Mechanics of Biomaterials
Presented byAndrian SueAMME4981/9981Semester 1, 2016
Lecture 7
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Last Week
β Using motion to find forces and moments in the body (inverse problems)
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This Week
β Using the forces and moments to determine the stresses
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Elastic Behaviour
Hookeβs Law (Uniaxial) π = πΈΟ΅β Strain is directly proportional to the stress (Youngβs modulus)
Hookeβs Law (General) [π] = [π][π]β Stress tensor [π]β Strain tensor [π]β Stiffness tensor [π]
[π] = [π])*[π] = [πΆ][π]β Compliance tensor [πΆ] = [π])*
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Stress Calculation
β Undeformed
β Cauchy Stress (True Stress) π = -.
β Nominal Stress (Engineering Stress)
π =ΞπΏπΏ =
π β πΏπΏ =
ππΏ β 1 β π =
ππΏ = 1 + π
π =πΉπ΄ =
πΏπππΏππ
πΉπ΄ =
πππΏπ΄
πΏππΉπ =
πdefπundef
πΏππΉπ =
π/ππ/πC
1π/πΏ
πΉπ =
πCπ1π π
β Deformed
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Elastic Constants
Youngβs Modulus, Eβ Relationship between tensile or compressive stress and strainβ Only applies for small strains (within the elastic range)
*Assume linear, elastic, isotropic material
Biomaterial E (GPa)*Cancellous bone 0.49Cortical bone 14.7Long bone - Femur 17.2Long bone - Humerus 17.2Long bone - Radius 18.6Long bone - Tibia 18.1Vertebrae - Cervical 0.23Vertebrae - Lumbar 0.16
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Elastic Constants
Poissonβs Ratio, Ξ½β Describes the lateral deformation in response to an axial load
π = βπlateralπaxial
aFF
l Density Ο
L ΞL
r R
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Elastic Constants
Shear Modulus (or Lameβs second constant), G, ΞΌβ Describes the relationship between applied torque and angle of
deformation
πΊ = π =ππΎ =
ShearStressShearStrain
Bulk Modulus, Kβ Describes the resistance to uniform compression (hydrostatic pressure)
πΎ = βΞππ = β
ΞπΞπ/π β βπ
ππππ
Lameβs first constant, Ξ»β Used to simplify the stiffness matrix in Hookeβs law
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Elastic Constants
β Youngβs Modulus, E πΈ = X(Z[\]X)[\X = [(*\_)(*)]_)
_ = 2πΊ(1 + π)
β Poissonβs Ratio, Ξ½ π = []([\X) =
[(Za)[) =
b]X β 1
β Shear Modulus, G, ΞΌ πΊ = [(*)]_)]_ = b
](*\_)
β Bulk Modulus, K πΎ = bZ(*)]_)
β Lameβs Constant, Ξ» π = ]X_*)]_ =
X(b)]X)ZX)b = b_
(*\_)(*)]_)
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Hookeβs Law: Tensor Representation
β Hookeβs Law: [π] = [πΆ][π] or [π] = [π][π]
β Stress Tensor: [π] =πdd πde πdfπed πee πefπfd πfe πff
or [π] =π** π*] π*Zπ]* π]] π]ZπZ* πZ] πZZ
β Strain Tensor: [π] =πdd πde πdfπed πee πefπfd πfe πff
or [π] =π** π*] π*Zπ]* π]] π]ZπZ* πZ] πZZ
β In this form, [π] and [π] are 2nd order tensorsβ In this form, [πΆ] and [π] are 4th order tensorsβ Too difficult to determine [πΆ] and [π]
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Hookeβs Law: Matrix Representation
β Hookeβs Law: {π} = [πΆ]{π}
π =
π*π]πZπiπjπk
=
π**π]]πZZ2π]Z2π*Z2π*]
[πΆ] =
πΆ** πΆ*] πΆ*ZπΆ]* πΆ]] πΆ]ZπΆZ* πΆZ] πΆZZπΆi* πΆi] πΆiZπΆj* πΆj] πΆjZπΆk* πΆk] πΆkZ
πΆ*i πΆ*j πΆ*kπΆ]i πΆ]j πΆ]kπΆZi πΆZj πΆZkπΆii πΆij πΆikπΆji πΆjj πΆjkπΆkj πΆkj πΆkk
π =
π*π]πZπiπjπk
=
π**π]]πZZπ]Zπ*Zπ*]
β In this form, {π} and {π} are 1st order vectorsβ In this form, [πΆ] is a 2nd order tensorβ Much easier to determine [πΆ]β This is called the Voigt notation β reduces the order of the symmetric tensor
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Constitutive Material Models
[πΆ] =
πΆ** πΆ*] πΆ*ZπΆ]* πΆ]] πΆ]ZπΆZ* πΆZ] πΆZZπΆi* πΆi] πΆiZπΆj* πΆj] πΆjZπΆk* πΆk] πΆkZ
πΆ*i πΆ*j πΆ*kπΆ]i πΆ]j πΆ]kπΆZi πΆZj πΆZkπΆii πΆij πΆikπΆji πΆjj πΆjkπΆkj πΆkj πΆkk
Constitutive Model Number of Independent Components in [C]
Anisotropy 21Orthotropy 9Transverse Isotropy 5Isotropy 2
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Anisotropy
β Most general form of Hookeβs lawβ 21 independent componentsβ Example: wood
{π} = [πΆ]{π}
π**π]]πZZ2π]Z2π*Z2π*]
=
πΆ** πΆ*] πΆ*ZπΆ]* πΆ]] πΆ]ZπΆZ* πΆZ] πΆZZπΆi* πΆi] πΆiZπΆj* πΆj] πΆjZπΆk* πΆk] πΆkZ
πΆ*i πΆ*j πΆ*kπΆ]i πΆ]j πΆ]kπΆZi πΆZj πΆZkπΆii πΆij πΆikπΆji πΆjj πΆjkπΆki πΆkj πΆkk
π**π]]πZZπ]Zπ*Zπ*]
β Symmetric matrix: πΆ*] = πΆ]*,πΆ*Z = πΆZ*,ππ‘π.
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Orthotropy
β Material possesses symmetry about three orthogonal planesβ 9 independent components
β 3 Youngβs moduli: πΈ*,πΈ],πΈZβ 3 Poissonβs ratios: π*] = π]*,π]Z = πZ], πZ* = π*Zβ 3 shear moduli: πΊ*],πΊ]Z, πΊZ*
β Example: cortical bone
π**π]]πZZ2π]Z2π*Z2π*]
=
1πΈ*
βπ*]πΈ*
βπ*ZπΈ*
0 0 0
βπ*]πΈ]
1πΈ]
βπ]ZπΈ]
0 0 0
βπ*ZπΈZ
βπ]ZπΈZ
1πΈZ
0 0 0
0 0 01πΊ]Z
0 0
0 0 0 01πΊZ*
0
0 0 0 0 01πΊ*]
π**π]]πZZπ]Zπ*Zπ*]
1
2
3
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Orthotropy
β Example: cortical bone
β Large variations in property values are not necessarily (although may possibly be) due to experimental error
Component ValuesE1 6.91β18.1 GPaE2 8.51β19.4 GPaE3 17.0β26.5 GPaG12 2.41β7.22 GPaG12 3.28β8.65 GPaG12 3.28β8.67 GPaΞ½ij 0.12β0.62
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Transverse Isotropy
β 5 independent componentsβ Youngβs moduli: πΈ* = πΈ], πΈZβ Poissonβs ratios: π*] = π]*, π]Z = πZ] = πZ* = π*Zβ Shear modulus: πΊ]Z = πΊZ*, πΊ*] =
bq](*\_qr)
β Example: skin
π**π]]πZZ2π]Z2π*Z2π*]
=
1πΈ*
βπ*]πΈ*
βπ*ZπΈ*
0 0 0
βπ*]πΈ*
1πΈ*
βπ*ZπΈ*
0 0 0
βπ*ZπΈZ
βπ*ZπΈZ
1πΈZ
0 0 0
0 0 01πΊZ*
0 0
0 0 0 01πΊZ*
0
0 0 0 0 02(1 + π*])
πΈ*
π**π]]πZZπ]Zπ*Zπ*]
1
23
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Isotropy
β 2 independent components
β Youngβs modulus: πΈ = πΈ* = πΈ] = πΈZβ Poissonβs ratio: π = π*] = π]Z = πZ*, πΊ = πΊ]Z = πΊZ* = πΊ*] =
b](*\_)
β Example: Ti-6Al-4V
π**π]]πZZ2π]Z2π*Z2π*]
=
1πΈ β
ππΈ β
ππΈ 0 0 0
βππΈ
1πΈ β
ππΈ 0 0 0
βππΈ β
ππΈ
1πΈ 0 0 0
0 0 02(1 + π)
πΈ 0 0
0 0 0 02(1 + π)
πΈ 0
0 0 0 0 02(1 + π)
πΈ
π**π]]πZZπ]Zπ*Zπ*]
1
2
3
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Hookeβs Law (Isotropic): Stress-Strain Relationship
π = π π β πtu = ππ‘π π πΏtu + 2ππtu
π‘π π = πdd + πee + πff πΏtu = x1πππ = π0πππ β π
πdd =b
*\_ *)]_[ 1 β π πdd + π πee + πff ]
πee =b
*\_ *)]_[ 1 β π πee + π πff + πdd ]
πff =b
*\_ *)]_[ 1 β π πff + π πdd + πee ]
πde =b
*\_πde
πef =b
*\_πef
πfd =b
*\_πfd
or
πdd = π πdd + πee + πff + 2πΊπddπee = π πdd + πee + πff + 2πΊπeeπff = π πdd + πee + πff + 2πΊπff
πde = 2πΊπdeπef = 2πΊπefπfd = 2πΊπfd
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Hookeβs Law (Isotropic): Strain-Stress Relationship
π = πΆ π β πtu =1 + ππΈ
πtu βππΈπ‘π π πΏtu
π‘π π = πdd + πee + πff πΏtu = x1πππ = π0πππ β π
πdd =*b[πdd β π πee + πff ]
πee =*b[πee β π πff + πdd ]
πff =*b[πff β π πdd + πee ]
πde =*\_b
πde
πef =*\_b
πef
πfd =*\_b
πfd
or
πdd =*b[πdd β π πee + πff ]
πee =*b[πee β π πff + πdd ]
πff =*b[πff β π πdd + πee ]
πde =*]Xπde
πef =*]Xπef
πfd =*]Xπfd
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Biomechanics Methods
There are three methods that can be used to determine the biomechanical responses to loads:
1. Analytical method (Mechanics of Solids 1 and 2)
2. Biomechanical experimentation (testing)
3. Numerical techniques (FEM)
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Analytical Method: General Case
π}}π~~πff2π~f2πf}2π}~
=
1πΈ}
βπ~}πΈ~
βπf}πΈf
0 0 0
βπ}~πΈ}
1πΈ~
βπf~πΈf
0 0 0
βπ}fπΈ}
βπ~fπΈ~
1πΈf
0 0 0
0 0 01πΊ~f
0 0
0 0 0 01πΊf}
0
0 0 0 0 01πΊ}~
π}}π~~πffπ~fπf}π}~
z (3)
y (2) x (1)
ez
et
en
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πff = β-οΏ½οΏ½
π}}π~~πff2π~f2πf}2π}~
=
1πΈ}
βπ~}πΈ~
βπf}πΈf
0 0 0
βπ}~πΈ}
1πΈ~
βπf~πΈf
0 0 0
βπ}fπΈ}
βπ~fπΈ~
1πΈf
0 0 0
0 0 01πΊ~f
0 0
0 0 0 01πΊf}
0
0 0 0 0 01πΊ}~
00πff000
=
βπf}πffπΈf
βπf~πffπΈfπffπΈf000
z (3)
y (2) x (1)
ez
et
en
Fz Fz
Analytical Method: Pure Axial Load
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Analytical Method: Pure Bending
πff = Β±οΏ½οΏ½οΏ½eοΏ½οΏ½οΏ½
πff = Β±οΏ½οΏ½οΏ½dοΏ½οΏ½οΏ½
z (3)
y (2) x (1)
ez
et
en
Mxx
z (3)
y (2) x (1)
ez
et
en
Myy
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Analytical Method: Eccentric Axial Load
Using the principle of superposition
πff = β-οΏ½οΏ½ Β±
οΏ½οΏ½οΏ½eοΏ½οΏ½οΏ½
Β± οΏ½οΏ½οΏ½dοΏ½οΏ½οΏ½
= πΉf β*οΏ½ Β±
eοΏ½eοΏ½οΏ½οΏ½Β± dοΏ½d
οΏ½οΏ½οΏ½
π = β -οΏ½
π = Β±οΏ½eοΏ½
z (3)
y (2) x (1)
ez
et
enFz Fz ( )y~,x~
x
y
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Example: Analytical Method
Determine the maximum compressive stress on the bone, given F=200N, M=10Nm, the outer diameter of the bone is do=5cm, and the inner diameter of the bone is di=3cm.
Using the principle of superposition:
π = βοΏ½eοΏ½ β
-οΏ½ [πΌ = οΏ½
i (ποΏ½i β πti), π΄ = π ποΏ½] β πt] ]
π = β *CΓC.C]jοΏ½οΏ½Γ C.C]jοΏ½)C.C*jοΏ½ β
]CCοΏ½Γ C.C]jr)C.C*jr
π = β1.095πππ
F FMM
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Biomechanical Experimentation: Femoral Testing
Three-pointBending
Four-pointBending
FemoralNeck Test
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Numerical Techniques: Bovine Femur Modelling
Bovine Femur Sample CT Scanning ScanIP Modelling
Angela Shi, 2010 (Thesis)
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Experimentation & Numerical Techniques: Bovine Femur
Specimen from bovine femur sample
in-vitro experimental setup
ScanCAD model
Angela Shi, 2010 (Thesis)
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Experimentation & Numerical Techniques: Bovine Femur
XFEM fracture analysis
Angela Shi, 2010 (Thesis)
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Numerical Techniques: Inhomogeneity of Bone
HU
E
pCEHU ΟΟ =ββ
Material relationAngela Shi, 2010 (Thesis)
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Experimentation & Numerical Techniques: Femur Fracture
β In-vitro test of cadaver model β eXtend FEM (XFEM) in Abaqus
Angela Shi, 2010 (Thesis)
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Numerical Techniques: Dental Prostheses
β Whole Jaw Model
β Partial Jaw Model
CT Image Segmentation Sectional Curves CAD Model FE Model
PDLMolar
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Numerical Techniques: Dental Prostheses
β 3 unit, all ceramic dental bridge
Solid Model Von Mises Stress
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Summary
β Mechanics modelsβ Elastic constants
β Constitutive material modelsβ Number of independent components required to describe the material
modelβ Biomechanics
β Determining the biomechanical response to loads through analytical methods, biomechanical experimentation, and numerical techniques
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