Topic 5: Bone Mechanics

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Topic 5 Bone Mechanics

• Bone is a hard connective tissue
• Forms rigid skeleton
• Yield strain is small lt 0.01
• Elastic modulus is high (18 GPa) compared with
  normal working stresses
• Stress-strain relation is linear in elastic range
• Strain-rate dependence of stress is minor in
  normal conditions
• Bone is frequently approximated as a linear
  (Hookean) elastic material

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Bone Anisotropy

• Bone is a composite
• mineral matrix
• collagen fibers
• Bone has organized microstructure
• lamellar (layered)
• Haversian (tubular)
• trabecular (spongy, fabric-like)
• Elastic moduli vary with type of loading
• tension compression
• bending shear
• Elastic moduli vary with orientation
• transverse vs. axial
• Bone is anisotropic
• requires more than two elastic constants

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Linear Elasticity Constitutive Law
In terms of the Strain Energy, W
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Elasticity Tensor, Cijkl

• fourth-order tensor of elastic constants
• 34 81 components
• symmetry conditions
• Tij Tji ekl elk ? 6x6 36 independent
  constants
• 2W Cijkleijekl Cklijekleij
• ? Cijkl Cklij ? leaves 21 independent constants
• simplest special case Isotropy

• l and m are the Lamé constants

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Isotropic Hookean SolidsTechnical Constants

• Measured from standard tests
• Uniaxial test
• Youngs modulus, E slope of the stress-strain
  curve
• Poisson ratio, n (-) ratio transverseaxial
  strain
• T11 Ee11 l(e11 e22 e33) 2me11
• l(e11 ne11 ne11) 2me11
• ? E l(1 2n) 2m
• T22 0 l(e11e22e33) 2me22
• l(e22/n e22 e22) 2me22

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Isotropic Hookean SolidsTechnical Constants

• Shear modulus, G half slope of the shear stress
  vs. shear strain curve
• For i ? j, Tij 2meij
• ? G m
• Bulk modulus, K mean stress s0 divided by
  volume change, D (dilatation)

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Stiffness Matrix cij
Represent the stress and strain tensors as column
matrices si T11, T22, T33, T23, T13,
T12T ei e11, e22, e33, 2e23, 2e13,
2e12T si cijej cij is the (6x6)
stiffness matrix e.g. for isotropic Hookean
materials
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Compliance Matrix sij
The inverse of the stiffness matrix ei sij
sj sij is the (6x6) compliance matrix e.g.
for isotropic Hookean solids, in terms of the
technical constants
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Orthotropy

• bone often assumed to be orthotropic
• different properties in the three mutually
  perpendicular directions 3 Young's moduli 3
  shear moduli 3 independent Poisson ratios
• ? 3 uniaxial tests and 3 plane shear tests
• structural axes of orthotropic symmetry are
  defined by bone microstructure
• Long bone structural axes
• (1) radial
• (2) circumferential
• (3) longitudinal
• As for isotropy, stiffness matrix has 12 non-zero
  components, but 9 independent

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Orthotropy Stiffness Matrix

• Technical constants
• 3 Young's moduli for uniaxial strain along each
  axis, Ei
• 6 Poisson ratios, nij for strain in the
  j-direction when loaded in the i-direction (i ?
  j)
• nijEj njiEi (no sum) leaving 3 independent
  Poisson ratios
• 3 shear moduli, Gij Gji for shear in the i-j
  plane

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Orthotropy Compliance Matrix
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Transverse Isotropy

• E1 and E2 are similar compared with E3
• Similarly, n31 and n32 are close compared with
  n21
• ? greater differences between axial and
  transverse directions than between radial and
  circumferential
• Transversely isotropic materials
• one preferred (fiber) axis, i.e. long axis of
  the bone
• in long bones, the "fibers" are the osteons
• isotropic properties in plane transverse to
  fibers
• stiffness matrix simplifies from 9 to 5
  independent constants
• c11c22
• c13c23
• c44c55
• c660.5(c11-c12)

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Technical Constants for Human Bone
From SC Cowin, Chapter 2 in Handbook of
Bioengineering, 1987
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Bone Growth and Remodeling

• Bone continually remodels
• growth, reinforcement, resorption
• depends on stress and strain
• There is an optimal range of stress for maximum
  strength
• understressed or overstressed bone can weaken
• stresses on fractured bone affect healing
• stress-dependent remodeling affects surgical
  implant and prosthesis design, e.g. fracture
  fixation plates, surgical screws, artificial
  joints
• 1978 radiographic evidence of bone resorption
  seen in 70 of total hip replacement patients

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Stress-Dependent Remodeling

• Osteoclasts - cells responsible for resorption
• Osteoblasts - cells responsible for growth
  (hypertrophy)
• compressive stress stimulates formation of new
  bone and is important for fracture healing
• loss of normal stress ? loss of calcium and
  reduced bone density
• Time scales
• remodeling - months/years
• fastest remodeling is due to change in mineral
  content
• healing - weeks
• growth/maturation - years

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Types of Bone Remodeling

• Two types of remodeling in bone
• 1. surface (external) remodeling
• change in bone shape and dimensions
• deposition on to or resorption of bone material
  from inner or outer surfaces
• 2. internal remodeling
• change in
• bulk density
• trabecular size
• orientation
• osteon size, etc.

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Functional Adaptation

• Principal of Functional Adaptation, Roux (1895)
• "the ability of organs (and cells, tissues and
  organisms) to adapt their capacity to function in
  response to altered demands by practice
• Functional adaptation in bone is remodeling of
  structure, geometry and mechanical properties in
  response to altered loading
• Related to the engineering concept of optimal
  design

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Some Optimal Design Principles

• Theory of Uniform Strength attempts to produce
  the same maximum normal stress (brittle material)
  or shear stress (ductile material) throughout the
  body for a specific loading
• Theory of Trajectorial Architecture
  concentrates material in the paths of force
  transmission, such as principal stress lines,
  e.g. fiber reinforcing of composite
  (kevlar-mylar) sails
• Principle of Maximum-Minimum Design maximize
  strength for minimum weight or cost
• These theories have been verified in many cases

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Stress Adaptation of Trabecular Bone

• G.H. von Meyers trabecular bone architecture in
  human femur (1867)

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Remodeling of Trabecular Bone Wolff's Law

• Wolff (1872) when loads are changed by trauma or
  change in activity, functional remodeling
  reorients bone trabeculae so they align with the
  new principal stress axes
• Wolff never actually proved this
• Wolff's law of bone transformation (1884)
  there is a perfect mathematical correspondence
  between the structure of cancellous bone in
  proximal femur and Culmanns trajectories
• Culmanns trajectories and other of Wolffs
  assertions were suspect, but photoelastic studies
  (Pauwels,1954) confirmed Wolff's law

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Equations of Linear Elastostatics
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Example Problem in Linear ElasticitySimple
Torsion of a Circular Shaft
See section 12.4 (pages 274-278 in YC Fung. A
First Course in Continuum Mechanics, 3rd Ed., 1994
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Simple Torsion of a Circular ShaftStress and
Strain Components
Strain components
Stress components (from the Constitutive Law)
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Simple Torsion of a Circular ShaftEquilibrium
and Boundary Conditions
Equilibrium Equations (no body forces)
Boundary Conditions
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Resultant Forces and Moments
Integrating stresses over cross-section to get
force resultants
Resultant Torsional Moment, M
G shear modulus
GJ torsional rigidity
a twist per unit length
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Bone Mechanics Key Points

• Under physiological loads, bone can be assumed
  Hookean elastic with a high elastic modulus
  (10-20 GPa)
• The microstructure of the bone composite makes
  the material response anisotropic.
• Compared with an isotropic Hookean elastic solid
  which has two independent technical constants,
  transversely isotropic linearly elastic solids
  have five independent elastic constants and
  orthotropic Hookean solids have nine.
• For human cortical bone orthotropy is a somewhat
  better assumption than transverse isotropy, but
  transverse isotropy is a much better
  approximation than isotropy.
• The equilibrium equations, together with the
  constitutive equation for linear elasticity and
  the strain-displacement relation give us Naviers
  equations of linear elastostatics.
• They are used to solve boundary value problems
  for bone.

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Bone growth and remodeling Summary of key points

• Historical principles
• Wolffs Law
• Functional adaptation (stress-adaptive
  remodeling)
• Types of bone remodeling
• internal remodeling
• changes of bone density (and hence strength and
  stiffness)
• changes of trabecular architecture
• external remodeling
• changes of bone geometry
• Remodeling laws
• strain-energy dependent density remodeling
  (Carter)
• strain-dependent surface remodeling (Cowin)
• stress-dependent fabric tensor remodeling
  (Cowin)