Mechanics of Biomaterials - University of Sydney
Transcript of Mechanics of Biomaterials - University of Sydney
The University of Sydney Slide 1
Mechanics of Biomaterials
Presented byAndrian SueAMME4981/9981Semester 1, 2016
Lecture 7
The University of Sydney Slide 3
Last Week
โ Using motion to find forces and moments in the body (inverse problems)
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This Week
โ Using the forces and moments to determine the stresses
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Elastic Behaviour
Hookeโs Law (Uniaxial) ๐ = ๐ธฯตโ Strain is directly proportional to the stress (Youngโs modulus)
Hookeโs Law (General) [๐] = [๐][๐]โ Stress tensor [๐]โ Strain tensor [๐]โ Stiffness tensor [๐]
[๐] = [๐])*[๐] = [๐ถ][๐]โ Compliance tensor [๐ถ] = [๐])*
The University of Sydney Slide 6
Stress Calculation
โ Undeformed
โ Cauchy Stress (True Stress) ๐ = -.
โ Nominal Stress (Engineering Stress)
๐ =ฮ๐ฟ๐ฟ =
๐ โ ๐ฟ๐ฟ =
๐๐ฟ โ 1 โ ๐ =
๐๐ฟ = 1 + ๐
๐ =๐น๐ด =
๐ฟ๐๐๐ฟ๐๐
๐น๐ด =
๐๐๐ฟ๐ด
๐ฟ๐๐น๐ =
๐def๐undef
๐ฟ๐๐น๐ =
๐/๐๐/๐C
1๐/๐ฟ
๐น๐ =
๐C๐1๐ ๐
โ Deformed
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Elastic Constants
Youngโs Modulus, Eโ Relationship between tensile or compressive stress and strainโ 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
The University of Sydney Slide 8
Elastic Constants
Poissonโs Ratio, ฮฝโ Describes the lateral deformation in response to an axial load
๐ = โ๐lateral๐axial
aFF
l Density ฯ
L ฮL
r R
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Elastic Constants
Shear Modulus (or Lameโs second constant), G, ฮผโ Describes the relationship between applied torque and angle of
deformation
๐บ = ๐ =๐๐พ =
ShearStressShearStrain
Bulk Modulus, Kโ Describes the resistance to uniform compression (hydrostatic pressure)
๐พ = โฮ๐๐ = โ
ฮ๐ฮ๐/๐ โ โ๐
๐๐๐๐
Lameโs first constant, ฮปโ Used to simplify the stiffness matrix in Hookeโs law
The University of Sydney Slide 10
Elastic Constants
โ Youngโs Modulus, E ๐ธ = X(Z[\]X)[\X = [(*\_)(*)]_)
_ = 2๐บ(1 + ๐)
โ Poissonโs Ratio, ฮฝ ๐ = []([\X) =
[(Za)[) =
b]X โ 1
โ Shear Modulus, G, ฮผ ๐บ = [(*)]_)]_ = b
](*\_)
โ Bulk Modulus, K ๐พ = bZ(*)]_)
โ Lameโs Constant, ฮป ๐ = ]X_*)]_ =
X(b)]X)ZX)b = b_
(*\_)(*)]_)
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Hookeโs Law: Tensor Representation
โ Hookeโs Law: [๐] = [๐ถ][๐] or [๐] = [๐][๐]
โ Stress Tensor: [๐] =๐dd ๐de ๐df๐ed ๐ee ๐ef๐fd ๐fe ๐ff
or [๐] =๐** ๐*] ๐*Z๐]* ๐]] ๐]Z๐Z* ๐Z] ๐ZZ
โ Strain Tensor: [๐] =๐dd ๐de ๐df๐ed ๐ee ๐ef๐fd ๐fe ๐ff
or [๐] =๐** ๐*] ๐*Z๐]* ๐]] ๐]Z๐Z* ๐Z] ๐ZZ
โ In this form, [๐] and [๐] are 2nd order tensorsโ In this form, [๐ถ] and [๐] are 4th order tensorsโ Too difficult to determine [๐ถ] and [๐]
The University of Sydney Slide 12
Hookeโs Law: Matrix Representation
โ Hookeโs Law: {๐} = [๐ถ]{๐}
๐ =
๐*๐]๐Z๐i๐j๐k
=
๐**๐]]๐ZZ2๐]Z2๐*Z2๐*]
[๐ถ] =
๐ถ** ๐ถ*] ๐ถ*Z๐ถ]* ๐ถ]] ๐ถ]Z๐ถZ* ๐ถZ] ๐ถZZ๐ถi* ๐ถi] ๐ถiZ๐ถj* ๐ถj] ๐ถjZ๐ถk* ๐ถk] ๐ถkZ
๐ถ*i ๐ถ*j ๐ถ*k๐ถ]i ๐ถ]j ๐ถ]k๐ถZi ๐ถZj ๐ถZk๐ถii ๐ถij ๐ถik๐ถji ๐ถjj ๐ถjk๐ถkj ๐ถkj ๐ถkk
๐ =
๐*๐]๐Z๐i๐j๐k
=
๐**๐]]๐ZZ๐]Z๐*Z๐*]
โ In this form, {๐} and {๐} are 1st order vectorsโ In this form, [๐ถ] is a 2nd order tensorโ Much easier to determine [๐ถ]โ This is called the Voigt notation โ reduces the order of the symmetric tensor
The University of Sydney Slide 14
Constitutive Material Models
[๐ถ] =
๐ถ** ๐ถ*] ๐ถ*Z๐ถ]* ๐ถ]] ๐ถ]Z๐ถZ* ๐ถZ] ๐ถZZ๐ถi* ๐ถi] ๐ถiZ๐ถj* ๐ถj] ๐ถjZ๐ถk* ๐ถk] ๐ถkZ
๐ถ*i ๐ถ*j ๐ถ*k๐ถ]i ๐ถ]j ๐ถ]k๐ถZi ๐ถZj ๐ถZk๐ถii ๐ถij ๐ถik๐ถji ๐ถjj ๐ถjk๐ถkj ๐ถkj ๐ถkk
Constitutive Model Number of Independent Components in [C]
Anisotropy 21Orthotropy 9Transverse Isotropy 5Isotropy 2
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Anisotropy
โ Most general form of Hookeโs lawโ 21 independent componentsโ Example: wood
{๐} = [๐ถ]{๐}
๐**๐]]๐ZZ2๐]Z2๐*Z2๐*]
=
๐ถ** ๐ถ*] ๐ถ*Z๐ถ]* ๐ถ]] ๐ถ]Z๐ถZ* ๐ถZ] ๐ถZZ๐ถi* ๐ถi] ๐ถiZ๐ถj* ๐ถj] ๐ถjZ๐ถk* ๐ถk] ๐ถkZ
๐ถ*i ๐ถ*j ๐ถ*k๐ถ]i ๐ถ]j ๐ถ]k๐ถZi ๐ถZj ๐ถZk๐ถii ๐ถij ๐ถik๐ถji ๐ถjj ๐ถjk๐ถki ๐ถkj ๐ถkk
๐**๐]]๐ZZ๐]Z๐*Z๐*]
โ Symmetric matrix: ๐ถ*] = ๐ถ]*,๐ถ*Z = ๐ถZ*,๐๐ก๐.
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Orthotropy
โ Material possesses symmetry about three orthogonal planesโ 9 independent components
โ 3 Youngโs moduli: ๐ธ*,๐ธ],๐ธZโ 3 Poissonโs ratios: ๐*] = ๐]*,๐]Z = ๐Z], ๐Z* = ๐*Zโ 3 shear moduli: ๐บ*],๐บ]Z, ๐บZ*
โ Example: cortical bone
๐**๐]]๐ZZ2๐]Z2๐*Z2๐*]
=
1๐ธ*
โ๐*]๐ธ*
โ๐*Z๐ธ*
0 0 0
โ๐*]๐ธ]
1๐ธ]
โ๐]Z๐ธ]
0 0 0
โ๐*Z๐ธZ
โ๐]Z๐ธZ
1๐ธZ
0 0 0
0 0 01๐บ]Z
0 0
0 0 0 01๐บZ*
0
0 0 0 0 01๐บ*]
๐**๐]]๐ZZ๐]Z๐*Z๐*]
1
2
3
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Orthotropy
โ Example: cortical bone
โ Large variations in property values are not necessarily (although may possibly be) due to experimental error
Component ValuesE1 6.91โ18.1 GPaE2 8.51โ19.4 GPaE3 17.0โ26.5 GPaG12 2.41โ7.22 GPaG12 3.28โ8.65 GPaG12 3.28โ8.67 GPaฮฝij 0.12โ0.62
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Transverse Isotropy
โ 5 independent componentsโ Youngโs moduli: ๐ธ* = ๐ธ], ๐ธZโ Poissonโs ratios: ๐*] = ๐]*, ๐]Z = ๐Z] = ๐Z* = ๐*Zโ Shear modulus: ๐บ]Z = ๐บZ*, ๐บ*] =
bq](*\_qr)
โ Example: skin
๐**๐]]๐ZZ2๐]Z2๐*Z2๐*]
=
1๐ธ*
โ๐*]๐ธ*
โ๐*Z๐ธ*
0 0 0
โ๐*]๐ธ*
1๐ธ*
โ๐*Z๐ธ*
0 0 0
โ๐*Z๐ธZ
โ๐*Z๐ธZ
1๐ธZ
0 0 0
0 0 01๐บZ*
0 0
0 0 0 01๐บZ*
0
0 0 0 0 02(1 + ๐*])
๐ธ*
๐**๐]]๐ZZ๐]Z๐*Z๐*]
1
23
The University of Sydney Slide 19
Isotropy
โ 2 independent components
โ Youngโs modulus: ๐ธ = ๐ธ* = ๐ธ] = ๐ธZโ Poissonโs ratio: ๐ = ๐*] = ๐]Z = ๐Z*, ๐บ = ๐บ]Z = ๐บZ* = ๐บ*] =
b](*\_)
โ Example: Ti-6Al-4V
๐**๐]]๐ZZ2๐]Z2๐*Z2๐*]
=
1๐ธ โ
๐๐ธ โ
๐๐ธ 0 0 0
โ๐๐ธ
1๐ธ โ
๐๐ธ 0 0 0
โ๐๐ธ โ
๐๐ธ
1๐ธ 0 0 0
0 0 02(1 + ๐)
๐ธ 0 0
0 0 0 02(1 + ๐)
๐ธ 0
0 0 0 0 02(1 + ๐)
๐ธ
๐**๐]]๐ZZ๐]Z๐*Z๐*]
1
2
3
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Hookeโs Law (Isotropic): Stress-Strain Relationship
๐ = ๐ ๐ โ ๐tu = ๐๐ก๐ ๐ ๐ฟtu + 2๐๐tu
๐ก๐ ๐ = ๐dd + ๐ee + ๐ff ๐ฟtu = x1๐๐๐ = ๐0๐๐๐ โ ๐
๐dd =b
*\_ *)]_[ 1 โ ๐ ๐dd + ๐ ๐ee + ๐ff ]
๐ee =b
*\_ *)]_[ 1 โ ๐ ๐ee + ๐ ๐ff + ๐dd ]
๐ff =b
*\_ *)]_[ 1 โ ๐ ๐ff + ๐ ๐dd + ๐ee ]
๐de =b
*\_๐de
๐ef =b
*\_๐ef
๐fd =b
*\_๐fd
or
๐dd = ๐ ๐dd + ๐ee + ๐ff + 2๐บ๐dd๐ee = ๐ ๐dd + ๐ee + ๐ff + 2๐บ๐ee๐ff = ๐ ๐dd + ๐ee + ๐ff + 2๐บ๐ff
๐de = 2๐บ๐de๐ef = 2๐บ๐ef๐fd = 2๐บ๐fd
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Hookeโs Law (Isotropic): Strain-Stress Relationship
๐ = ๐ถ ๐ โ ๐tu =1 + ๐๐ธ
๐tu โ๐๐ธ๐ก๐ ๐ ๐ฟtu
๐ก๐ ๐ = ๐dd + ๐ee + ๐ff ๐ฟtu = x1๐๐๐ = ๐0๐๐๐ โ ๐
๐dd =*b[๐dd โ ๐ ๐ee + ๐ff ]
๐ee =*b[๐ee โ ๐ ๐ff + ๐dd ]
๐ff =*b[๐ff โ ๐ ๐dd + ๐ee ]
๐de =*\_b
๐de
๐ef =*\_b
๐ef
๐fd =*\_b
๐fd
or
๐dd =*b[๐dd โ ๐ ๐ee + ๐ff ]
๐ee =*b[๐ee โ ๐ ๐ff + ๐dd ]
๐ff =*b[๐ff โ ๐ ๐dd + ๐ee ]
๐de =*]X๐de
๐ef =*]X๐ef
๐fd =*]X๐fd
The University of Sydney Slide 23
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)
The University of Sydney Slide 24
Analytical Method: General Case
๐}}๐~~๐ff2๐~f2๐f}2๐}~
=
1๐ธ}
โ๐~}๐ธ~
โ๐f}๐ธf
0 0 0
โ๐}~๐ธ}
1๐ธ~
โ๐f~๐ธf
0 0 0
โ๐}f๐ธ}
โ๐~f๐ธ~
1๐ธf
0 0 0
0 0 01๐บ~f
0 0
0 0 0 01๐บf}
0
0 0 0 0 01๐บ}~
๐}}๐~~๐ff๐~f๐f}๐}~
z (3)
y (2) x (1)
ez
et
en
The University of Sydney Slide 25
๐ff = โ-๏ฟฝ๏ฟฝ
๐}}๐~~๐ff2๐~f2๐f}2๐}~
=
1๐ธ}
โ๐~}๐ธ~
โ๐f}๐ธf
0 0 0
โ๐}~๐ธ}
1๐ธ~
โ๐f~๐ธf
0 0 0
โ๐}f๐ธ}
โ๐~f๐ธ~
1๐ธf
0 0 0
0 0 01๐บ~f
0 0
0 0 0 01๐บf}
0
0 0 0 0 01๐บ}~
00๐ff000
=
โ๐f}๐ff๐ธf
โ๐f~๐ff๐ธf๐ff๐ธf000
z (3)
y (2) x (1)
ez
et
en
Fz Fz
Analytical Method: Pure Axial Load
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Analytical Method: Pure Bending
๐ff = ยฑ๏ฟฝ๏ฟฝ๏ฟฝe๏ฟฝ๏ฟฝ๏ฟฝ
๐ff = ยฑ๏ฟฝ๏ฟฝ๏ฟฝd๏ฟฝ๏ฟฝ๏ฟฝ
z (3)
y (2) x (1)
ez
et
en
Mxx
z (3)
y (2) x (1)
ez
et
en
Myy
The University of Sydney Slide 27
Analytical Method: Eccentric Axial Load
Using the principle of superposition
๐ff = โ-๏ฟฝ๏ฟฝ ยฑ
๏ฟฝ๏ฟฝ๏ฟฝe๏ฟฝ๏ฟฝ๏ฟฝ
ยฑ ๏ฟฝ๏ฟฝ๏ฟฝd๏ฟฝ๏ฟฝ๏ฟฝ
= ๐นf โ*๏ฟฝ ยฑ
e๏ฟฝe๏ฟฝ๏ฟฝ๏ฟฝยฑ d๏ฟฝd
๏ฟฝ๏ฟฝ๏ฟฝ
๐ = โ -๏ฟฝ
๐ = ยฑ๏ฟฝe๏ฟฝ
z (3)
y (2) x (1)
ez
et
enFz Fz ( )y~,x~
x
y
The University of Sydney Slide 28
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:
๐ = โ๏ฟฝe๏ฟฝ โ
-๏ฟฝ [๐ผ = ๏ฟฝ
i (๐๏ฟฝi โ ๐ti), ๐ด = ๐ ๐๏ฟฝ] โ ๐t] ]
๐ = โ *CรC.C]j๏ฟฝ๏ฟฝร C.C]j๏ฟฝ)C.C*j๏ฟฝ โ
]CC๏ฟฝร C.C]jr)C.C*jr
๐ = โ1.095๐๐๐
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)
The University of Sydney Slide 31
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 ฯฯ =โโ
Material relationAngela Shi, 2010 (Thesis)
The University of Sydney Slide 34
Experimentation & Numerical Techniques: Femur Fracture
โ In-vitro test of cadaver model โ eXtend FEM (XFEM) in Abaqus
Angela Shi, 2010 (Thesis)
The University of Sydney Slide 35
Numerical Techniques: Dental Prostheses
โ Whole Jaw Model
โ Partial Jaw Model
CT Image Segmentation Sectional Curves CAD Model FE Model
PDLMolar
The University of Sydney Slide 36
Numerical Techniques: Dental Prostheses
โ 3 unit, all ceramic dental bridge
Solid Model Von Mises Stress
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Summary
โ Mechanics modelsโ Elastic constants
โ Constitutive material modelsโ Number of independent components required to describe the material
modelโ Biomechanics
โ Determining the biomechanical response to loads through analytical methods, biomechanical experimentation, and numerical techniques