5.7 Energy and its relation to material response

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

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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 ($W_F$)

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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
  • …

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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

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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)

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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

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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)

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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