6 Stress and strain

6.1 Stress

6.1.1 Normal stress

Axial member

Axial member

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

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

6.1.2 Shear stress

Shear member

Shear member

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

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

6.1.3 Stresses on arbitrary planes

tractionvectors

  • Since different “cuts” must yield the same resultant force, the stress depends on your plane of observation
  • Each type of stress is simultaneously present13
    • 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

6.1.4 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

6.2 Strain

6.2.1 Normal strain

Normal strain

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

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

6.2.2 Shear strain

Shear strain

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

6.2.3 Strain

Note

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

6.2.4 A chain of relationships in biomechanics

Constraints
\(\Updownarrow\)
Deformation
\(\Updownarrow\)
Strain
\(\Updownarrow\)
Stress
\(\Updownarrow\)
Equilibrium
\(\Updownarrow\)
Applied Loads
  • Each arrow represents a relationship that must be understood and properly accounted

  1. except in unusual circumstances↩︎