7 Skeletal structures

7.1 Deformation and stiffness

7.1.1 Skeletal structures and types of load

  • Bending
  • Axial Loading
    • Tension
    • Compression
  • Torsion

7.1.2 Stiffness

  • Stiffness is a structural relationship between deflection and load

(85,18)\(F\) (35,40)\(x\)

(50,-10)\(x\) (-10,40)\(F\) (60,40)\(k\)

  • The slope of the load-deflection curve (\(k\)) is the structural stiffness

  • Two factors influence the stiffness

    • Material response

    • Geometry

7.1.3 Structural properties

Material Properties

  • Elastic modulus (\(E\) and \(G\))

  • Yield stress (\(\sigma_Y\))

  • Toughness (brittle/ductile) (\(K\))

Independent of shape!

Geometric Properties

  • Material distribution

    • Cross sectional area (\(A\))

    • Area moment of inertia (\(I\))

    • Polar moment of inertia (\(J\))

Independent of material!

Structural Properties

  • Axial Stiffness (\(E A\))

  • Bending Stiffness (\(E I\))

  • Torsional Stiffness (\(G J\))

Depends on shape and material!

7.2 Material properties

7.2.1 Stress \(\Longleftrightarrow\) Strain

7.2.1.1 \(E\) – the elastic modulus

  • The elastic modulus is the most critical material property

(50,-10)Strain \(\varepsilon = \frac{\Delta l}{l}\) (-50,40)Stress \(\sigma = \frac{P}{A}\) (60,40)Slope – \(E\)

7.2.2 Stress \(\Longleftrightarrow\) Strain

  • Moduli for common materials (GPa)
Stainless Steel 200
Titanium 110
Cortical Bone 7-21
Bone Cement 2.5-3.5
Cancellous Bone 0.7-4.9
UHMW-PE 1.4-4.2

(50,-10)\(\varepsilon\) (-10,40)\(\sigma\) (60,40)Slope –\(E\)

7.3 Energy and its relation to material response

7.3.1 Elastic-plastic behavior

(50,-10)\(\varepsilon\) (-10,40)\(\sigma\) (19,25)\(E\) (25,10)\(W_P\) (47,10)\(W_F\) (10,68)\(\sigma_Y\) (95,88)\(\sigma_{UTS}\)

  • Initial loading is “elastic” (no permanent deformation)

    • unloads back to origin

  • Elastic energy is stored in the material

    • it can be recovered (like a spring)

      • \(U=\frac{1}{2} \sigma \varepsilon = \frac{1}{2} E \varepsilon^2\)

7.3.2 Elastic-plastic behavior

(50,-10)\(\varepsilon\) (-10,40)\(\sigma\) (19,25)\(E\) (25,10)\(W_P\) (47,10)\(W_F\) (10,68)\(\sigma_Y\) (95,88)\(\sigma_{UTS}\)

  • Loading past yield (\(\sigma_Y\)) causes permanent set

    • Typical unload follows the slope of the elastic region

    • Energy is dissipated as plastic work (\(W_P\))

  • Loading to the ultimate tensile strength (\(\sigma_{UTS}\)) causes failure and additional energy dissipation (\(W_F\))

7.3.3 Bone density and the elastic modulus

  • Bone density a strong effect on modulus and other properties

  • Subtle changes greatly changes strength and elastic modulus

  • Density changes from:

    • normal aging

    • disease

    • malnutrition

    • use

    • disuse

    • ...

7.3.4 Energy and energy dissipation

  • In orthopaedics, two kinds of energy are of great concern:

    • kinetic and potential

  • Kinetic energy is the energy of a particle in motion \[K = \frac{1}{2} m V^2\]

    • Examples: gun shot impact, motor vehicle crash

  • Potential energy is the energy associated with a fall from a height \[U = m g h\]

    • During the fall, all potential energy is converted to kinetic energy just before impact

    • There are other relevant forms of potential energy

7.3.5 Energy and energy dissipation

  • Energy is “conserved”, all energy in the system goes to something

  • If enough energy is available, some goes to permanent deformation of the “structure” (bones, soft tissue, implants, etc)

7.3.6 Toughness: brittle vs ductile

(50,-10)\(\varepsilon\) (-10,40)\(\sigma\) (30,80)Brittle (90,80)Ductile

  • The fracture toughness is a measure of energy required to propagate a crack through a material

  • Brittle materials have low toughness, not much energy is required

  • <.-> Ductile materials have high toughness, much energy is required

7.3.7 Strength vs toughness

(50,-10)\(\varepsilon\) (-10,40)\(\sigma\) (45,90)Strong, brittle (90,76)Weak, ductile

  • Strength (\(\sigma_{UTS}\)) is a measure of how much
    stress a material can carry

  • Toughness is a measure of energy dissipated
    during failure (crack propagation)

7.3.8 Strength vs fatigue strength

(40,-10)\(\log N_f\) (-30,40)\(\sigma_{\mathrm{cylic}}\) (10,85)\(\sigma_{UTS}\) (100,32)\(\sigma_f\)

  • Cyclic loading (repetitive load and unload) can cause “fatigue failures” at loads much lower than the ultimate tensile strength

  • The S-N curve – plot of load vs number of cycles to failure

  • Some materials exhibit a fatigue strength (\(\sigma_f\))

    • The curve levels off and the material has infinite fatigue life below that stress

7.4 Properties of bone

7.4.1 Mechanics of bone: anisotropy

  • Isotropy –

    • Most metals – stainless, titanium, cobalt chrome

  • Anisotropy –

    • Strength and modulus both depend on direction

    • Bone is weakest in shear, then tension, then compression.

Ultimate stress at failure cortical bone
Compression \(< 212\) N/m\(^2\)
Tension \(< 146\) N/m\(^2\)
Shear \(< 82\) N/m\(^2\)

7.4.2 Types of bone fracture

7.4.3 Types of bone fracture

7.4.3.1 Axial load

image

  • In tension, failure occurs due to normal stress

  • In compression, failure occurs on the plane where shear is maximized

7.4.4 Types of bone fracture

7.4.4.1 Bending load

  • Compression strength is greater than tensile strength

  • Fails in tension, possibly with a butterfly fragment

7.4.5 Types of bone fracture

7.4.5.1 Bending and compression load

  • Combined compression and bending leads to
    oblique fracture with butterfly fragmentl

7.4.6 Types of bone fracture

7.4.6.1 Torsion

  • Like stress, components of strain depend
    on direction of observation

  • When torsion applied, tension occurs on a diagonal

  • Fractures propagate perpendicular to this tension diagonal

  • Spiral fracture 45\(^o\) to the long axis

7.5 Geometric properties

7.5.1 Properties of a cross section

Structural Properties

  • Axial Stiffness (\(E A\))

  • Bending Stiffness (\(E I\))

  • Torsional Stiffness (\(G J\))

Depends on shape and material!

  • We have seen that there is a geometric component to stiffness

  • We will examine three properties which are critical

    • Cross sectional area (\(A\))

    • Area moment of inertia (\(I\))

    • Polar (torsional) moment of inertia (\(J\))

7.5.2 Properties of a cross section

(99,50)\(h\) (35,-15)\(b\)

The area is critical in axial load because the stress is inversely proportional to the area \[\sigma = \frac{P}{A} = \frac{P}{b \cdot h}\] Axial stiffness (\(EA\)) is proportional to the area

7.5.3 Properties of a cross section

(99,50)\(h\) (35,-15)\(b\)

  • The area moment of inertia (\(I\)) goes as the third power of the thickness (about the axis of bending) \[I = \frac{b h^3}{12}\]

  • The bending stiffness (\(EI\)) is directly proportional to \(I\)

  • Additionally, \(h\) also critical in bending because of its relation the maximum stress

\[\sigma_{\mathrm{max}} = \frac{M h}{2 I} = \frac{6 M}{b h^2}\]

7.5.4 Properties of a cross section

(35,22)\(r\) (35,6)\(R\)

Similarly, the area moment of inertia for a hollow bone is: \[I = \frac{\pi (R^4-r^4)}{4}\]

Consider an application: IM Nails

Kolossos, via Wikipedia, Creative Commons Attribution-Share Alike 3.0

7.5.5 Implications for a fracture callus

  • As the callus increases the radius, the stiffness increases by \(R^4\)

  • The stress (for the same load) reduces by \(\frac{1}{R^3}\)

  • (These equations for circular cross sections)

7.5.6 Stiffness as a function of healing time

  • Callus increases with time

  • Stiffness increases with time

  • Near normal stiffness at 27 days

  • Does not correspond to radiographs

Figure from: Browner et al, Skeletal Trauma, 2nd Ed, Saunders, 1998.

7.5.7 IM Nail Diameter

Figure from: Tencer et al, Biomechanics in Orthopaedic Trauma, Lippincott, 1994.

7.5.8 Slotting

  • Results in more flexibility in bending and torsion

  • Decreases torsional strength by significant amount

Figure from: Tencer et al, Biomechanics in Orthopaedic Trauma, Lippincott, 1994.

\[GJ_{\mathrm{open}} = G \frac{b t^3}{3} = G \frac{\pi d t^3}{3}\] \[GJ_{\mathrm{ef}}^{\mathrm{cir}}=G \frac{\pi d^{3} t}{4}\]

Assume: \[t=3 \mathrm{mm}\] \[d=25 \mathrm{mm}\]

The ratio is: \[\frac{J_{\mathrm{ef}}^{\mathrm{cir}}}{J_{\mathrm{open}}}=52\]

7.5.9 Mechanics of bone: viscoelasticity

(50,-10)\(\varepsilon\) (-10,40)\(\sigma\) (58,56)\(\dot{\varepsilon}\) increasing

  • Bone is viscoelastic: its stress-strain characteristics are dependent upon the rate of loading

  • Example: trabecular bone becomes stiffer in compression the faster it is loaded.

7.5.10 Credits

  • Orthopaedic Basic Science, Einhorn, O’Keefe, and Buckwalter