Summarizing Elasticity: Stress and Strain

Summary

• Hooke’s law is given by
  \(F=k\text{Δ}L,\)

  where \(\Delta L\) is the amount of deformation (the change in length), \(F\) is the applied force, and \(k\) is a proportionality constant that depends on the shape and composition of the object and the direction of the force. The relationship between the deformation and the applied force can also be written as

  \(\Delta L=\frac{1}{Y}\frac{F}{A}{L}_{0},\)

  where \(Y\phantom{\rule{0.25em}{0ex}}\) is Young’s modulus, which depends on the substance, \(A\) is the cross-sectional area, and \({L}_{0}\) is the original length.

• The ratio of force to area, \(\frac{F}{A}\), is defined as stress, measured in N/m².
• The ratio of the change in length to length, \(\frac{\Delta L}{{L}_{0}}\), is defined as strain (a unitless quantity). In other words,
  \(\text{stress}=Y×\text{strain}.\)
• The expression for shear deformation is
  \(\Delta x=\frac{1}{S}\frac{F}{A}{L}_{0},\)

  where \(S\) is the shear modulus and \(F\) is the force applied perpendicular to \({L}_{\text{0}}\) and parallel to the cross-sectional area \(A\).

• The relationship of the change in volume to other physical quantities is given by
  \(\Delta V=\frac{1}{B}\frac{F}{A}{V}_{0},\)

  where \(B\) is the bulk modulus, \({V}_{\text{0}}\) is the original volume, and \(\frac{F}{A}\) is the force per unit area applied uniformly inward on all surfaces.

Glossary

deformation

change in shape due to the application of force

Hooke’s law

proportional relationship between the force \(F\) on a material and the deformation \(\Delta L\) it causes, \(F=k\Delta L\)

tensile strength

the breaking stress that will cause permanent deformation or fraction of a material

stress

ratio of force to area

strain

ratio of change in length to original length

shear deformation

deformation perpendicular to the original length of an object