Tensile stress
A force acting in a direction perpendicular to the surface it displaces will cause tension. The stress and strain in this case are called tensile stress and tensile strain, respectively, whilst the name of the corresponding elastic modulus is Young’s modulus.
Tensile strain
Tensile strain is the ratio of the change \(\Delta L\) in a dimension to the original value \(L_0\) of the same dimension:
\[\epsilon := \frac{\Delta L}{L_0}.\]
Tensile strain, as a ratio of quantities having the same unit, is unitless.
Tensile stress
Tensile stress is the ratio of the force \(F\) causing the object to get deformed to the area \(A\) of the surface being displaced:
\[\sigma := \frac{F}{A}.\]
The SI unit of tensile stress is the newton per square metre, also called the pascal:
\[\left[\sigma\right] = 1\,\frac{\mathrm{N}}{\mathrm{m}^2} = 1\,\mathrm{Pa}.\]
Hooke’s law
The tensile stress \(\sigma\) required to extend or compress an object is directly proportional the resulting strain \(\epsilon.\) The constant of proportionality is Young’s modulus \(Y.\)
\[\sigma = Y \epsilon.\]
The SI unit of Young’s modulus is the newton per square metre, also called the pascal:
\[\left[Y\right] = 1\,\frac{\mathrm{N}}{\mathrm{m}^2} = 1\,\mathrm{Pa}.\]
Hooke’s law for springs
The force law for springs \(F = -k \Delta L\) is also called Hooke’s law. We can easily see that this is essentially the same law as the one above:
\[\sigma = Y \epsilon,\]
\[\frac{F}{A} = Y \frac{\Delta L}{L_0},\]
\[F = \frac{A Y}{L_0} \Delta L.\]
For small deformations, we can assume that the cross-section area \(A\) of the spring stays constant, and thus the term \(\frac{A Y}{L_0}\) can be combined into a single spring constant \(k\):
\[k := \frac{A Y}{L_0},\]
\[F = k \Delta L.\]