5.7 Energy and its relation to material response

5.7.1 Elastic-plastic behavior

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

5.7.2 Elastic-plastic behavior

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  • 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 (\(W_F\))

5.7.3 Bone density and the elastic modulus

@Browner1998

  • 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
    • …

5.7.4 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

5.7.5 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)

5.7.6 Toughness: brittle vs ductile

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  • 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

5.7.7 Strength vs toughness

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  • 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)

5.7.8 Strength vs fatigue strength

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  • 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