Forces, moments, and equilibrium

Forces

Note Mass (\(m\)) can be considered the bodies resistance to a change in its motion (ie inertia))

Moments

Notes

• Moments are also called bending moments or torques
  depending on the application.

• Moments and forces are both described as “generalized forces”

Equilibrium

• We have already used another important concept, equilibrium

• Forces and changes in momentum remain in balance \[\begin{split} \displaystyle\sum F =& \; m a \\ \displaystyle\sum M =& \; I \alpha \\ \end{split}\]

• Much of biomechanics boils down to applying these equations

• Static equilibrium – the balance of forces that occurs
  when there is no acceleration \[\begin{split} \sum F =& \; 0 \\ \sum M =& \; 0 \\ \end{split}\]

“Mathematical tools” in biomechanics

Vectors and scalars

Vectors

Vector sums

• Vector addition is component by component
• Proper (2D) summation results in a parallelogram

Scalars

• Scalars have magnitude without direction.
• Examples
  • Levels of gray in a image
  • Patient mass
  • Patient bone density
  • Concentration of a drug per unit volume
  • Time
  • Altitude

Relationship between vectors and scalars

• Scalars and vectors can be linked mathematically

Example: diffusion of nutrients into intervertebral disk driven by gradient of glucose

Gradient of a scalar

\[\nabla f = \left(\frac{\partial f}{\partial x_1 }, \dots, \frac{\partial f}{\partial x_n } \right).\]

Gradient of a vector becomes a 2nd rank tensor \[\mathbf{f}=({{f}_{1}},{{f}_{2}},{{f}_{3}})\] \[\nabla \mathbf{f}=\frac{\partial {{f}_{j}}}{\partial {{x}_{i}}}{{\mathbf{e}}_{i}}{{\mathbf{e}}_{j}}\]

Example: strain is gradient of displacement vector

• \[\vec{u} = u {{e_x}}+ v {{e_y}}+ w {{e_z}}\]
• \[= {\varepsilon_{ij}}= \nabla \vec{u} = \left[ \begin{array}{ccc} {\varepsilon_{xx}}& {\varepsilon_{xy}}& {\varepsilon_{xz}}\\ {\varepsilon_{yx}}& {\varepsilon_{yy}}& {\varepsilon_{yz}}\\ {\varepsilon_{zx}}& {\varepsilon_{yz}}& {\varepsilon_{zz}}\\ \end{array} \right]\]

Divergence

• Similarly, the divergence operator (\(\nabla \cdot ()\)) reduces the
  rank of a tensor

  • i.e., makes a scalar out of a vector

• Example: the divergence calculates the strength
  of a source or sink of a velocity field in fluid flow

Rigid body and flexible body assumptions

• All bodies are flexible, meaning, all bodies deform when loaded
• For convenience, we often assume that a body is rigid
  • (ie we assume that it does not deform when loaded.)
• With this assumption, the mechanics/mathematics is simplified

Review of Stress and strain

Normal stress

• Normal stress is the resultant normal force over a given area

\[\sigma = \frac{P}{A}\]

Shear stress

• Shear stress is the resultant shear force over a given area

\[\tau = \frac{V}{A}\]

Stresses on arbitrary planes

• Since different “cuts” must yield the same resultant force, the stress depends on your plane of observation
• Each type of stress is simultaneously present¹
  • A body can fail in shear even when loaded by normal stress
  • Ductile materials typically yield due to shear stress
  • Brittle materials typically crack due to normal stress

Stress

Note

• Stresses result from equilibrium (ie the sum of forces)
• It is possible to have stress without strain
  • Example: thermal expansion/contraction
    • Exothermic reactions such as bone cement
    • Cement then adjusts to body temperature
    • Constrained by bone and implant \(\rightarrow\) stress

Strain

Normal strain

• Normal strain is the change in length over the original length

\[{\varepsilon}= \frac{\Delta l}{l}\]

Shear strain

• Shear strain (\(\gamma\)) is proportional to the shear angle (\(\alpha\)) \[\gamma = \alpha\]

Strain

Note

• Strain is defined by deformation
• It is possible to have strain without stress
  • Tissues expand with moisture content
  • PMMA shrinkage during polymerization

A chain of relationships in biomechanics

• Each arrow represents a relationship that must be understood and properly accounted

Deformation and stiffness

Skeletal structures and types of load

• Bending
• Axial Loading
  • Tension
  • Compression
• Torsion

Stiffness

• Stiffness is a structural relationship between deflection and load

• The slope of the load-deflection curve (\(k\)) is the structural stiffness
• Two factors influence the stiffness
  • Material response
  • Geometry

Structural properties

Structural Properties

• Axial Stiffness (\(E A\))
• Bending Stiffness (\(E I\))
• Torsional Stiffness (\(G J\))

Depends on shape and material!

Material properties

Stress \(\Longleftrightarrow\) Strain

\(E\) – the elastic modulus

• The elastic modulus is the most critical material property

• Strain \(\varepsilon = \frac{\Delta l}{l}\)
• Stress \(\sigma = \frac{P}{A}\)
• Slope – \(E\)

Stress \(\Longleftrightarrow\) Strain

• Moduli for common materials (GPa)

Energy and its relation to material response

Elastic-plastic behavior

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

Elastic-plastic behavior

• 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 (\(W_F\))

Bone density and the elastic modulus

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

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)

Toughness: brittle vs 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

Strength vs toughness

• 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)

Strength vs fatigue strength

• 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

Properties of bone

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.

Types of bone fracture

Types of bone fracture

Axial load

• In tension, failure occurs due to normal stress
• In compression, failure occurs on the plane where shear is maximized

Types of bone fracture

Bending load

• Compression strength is greater than tensile strength
• Fails in tension, possibly with a butterfly fragment

Types of bone fracture

Bending and compression load

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

Types of bone fracture

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

Geometric properties

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 (\(J\))

Properties of a cross section

• 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

Properties of a cross section

• 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

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

Properties of a cross section

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

Consider an application: IM Nails

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)

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

IM Nail Diameter

Slotting

• Results in more flexibility in bending and torsion
• Decreases torsional strength by significant amount

Equal values of mean width, equal thickness

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

Equal values of mean width, equal thickness

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

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

Mechanics of bone: viscoelasticity

• 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.

Credits

• @Einhorn2007
• @Le2004