A linear relationship exists between the strain ( ϵ ) imposed upon a material type and the corresponding stress ( σ ) needed to cause such a deformation; well, up to a limit :
The linear portion of the slope corresponds to magnitudes of strain that will cause a material to flex and snap back into its original configuration. The upper limit of this force is the yield strength, which is measured in Newtons ( N ) per square meter ( m2 ). Beyond this point, further strain will cause permanent yet somewhat gradual disfigurement to occur until the tensile strength or ultimate strength is attained. Further strain leads to the breaking strength being attained. Prior to reaching this point, disfigurement becomes extreme, and non-uniform deformation occurs until a complete breakdown of the forces keeping the material together is observed.
Q: A human femur is a hollow cylinder with an outer radius ( ro ) of 1.1 cm and an inner radius ( ri ) of 0.5 cm. If the compressive strength ( σc ) of bone is 170 MPa, how much force ( F ) is required to cause a rupture to occur?
A: We begin with the formula for compressive strain:
σc = ( F / A )
F = σcA
F = ( 170 x 106 Pa )( A )
A = 𝝿r2
A = ( 𝝿 )( ro2 – ri2 )
A = ( 𝝿 )[ ( 0.011 m )2 – ( 0.005 )2 ]
A = ( 𝝿 )( 0.000121 m2 – 0.000025 m2 )
A = ( 𝝿 )( 0.000096 m2 )
A = ( 3.1416 )( 0.000096 m2 )
A = 0.000301 m2 = 3.01 x 10-4 m2
F = ( 170 x 106 Pa )( 3.01 x 10-4 m2 )
F = 5.13 x 104 N
RENAISSANCE PHYSICS 