Materials

Hooke's law, stress-strain analysis, Young modulus and the mechanical properties of materials.

Spec Points Covered
  • I can state and apply Hooke's law, including identifying the limit of proportionality on a force-extension graph.
  • I can calculate the spring constantThe force per unit extension of a spring. A measure of the stiffness of the spring. Measured in N m⁻¹. from the gradient of a force-extension graph.
  • I can calculate the elastic potential energyThe capacity to do work. Measured in joules (J).The energyThe capacity to do work. Measured in joules (J). stored in a stretched or compressed spring (or other elastic object). stored in a stretched spring using $E = 1/2 kx^2$.
  • I can derive the combined spring constantThe force per unit extension of a spring. A measure of the stiffness of the spring. Measured in N m⁻¹. for springs in series and in parallel.
  • I can define and calculate stressThe force applied per unit cross-sectional area of a material. Measured in pascals (Pa)., strainThe fractional change in length of a material under stress. It is dimensionless (no units). and the Young modulusThe ratio of stressThe force applied per unit cross-sectional area of a material. Measured in pascals (Pa). to strainThe fractional change in length of a material under stress. It is dimensionless (no units). for a material in the elastic region. A measure of stiffness. Measured in pascals (Pa)..
  • I can interpret stressThe force applied per unit cross-sectional area of a material. Measured in pascals (Pa).-strainThe fractional change in length of a material under stress. It is dimensionless (no units). graphs and identify key features: limit of proportionality, elastic limit, yield point, UTS and fracture.
  • I can distinguish between elastic and plastic deformationPermanent deformation in which the material does not return to its original shape when the deforming force is removed. and explain the energyThe capacity to do work. Measured in joules (J). implications of each.
  • I can classify materials as brittle, ductile or polymeric from their stress-strain curves.
  • I can describe an experiment to measure the Young modulusThe ratio of stress to strain for a material in the elastic region. A measure of stiffness. Measured in pascals (Pa). of a wire.
Notes
01 Hooke's law Hooke's law → 02 Springs in parallel → 03 Elastic potential energy Elastic potential energy → 04 Stress Stress → 05 Limit of proportionality (P) → 06 Elastic deformation Elastic deformation → 07 Ductile materials (e → 08 Use a long →
Σ Key Equations Full Reference →
On Data Sheet
Not on Data Sheet
Hooke's law
$$F = kx$$
  • Where:
    • $F$ = N
    • $k$ = N \(m^{-1}\)
    • $x$ = m
Only valid up to the limit of proportionality. k is the spring constant (stiffness).
Elastic potential energy (spring constant form)
$$E = \frac{1}{2}kx^{2}$$
  • Where:
    • $E$ = J
    • $k$ = N \(m^{-1}\)
    • $x$ = m
Use when spring constant and extension are known. Only valid in the Hooke's law region.
Elastic potential energy (force form)
$$E = \frac{1}{2}Fx$$
  • Where:
    • $E$ = J
    • $F$ = N
    • $x$ = m
Equivalent to 1/2 kx^2 when F = kx. Area of the triangle under a linear force-extension graph.
Stress
$$\sigma = \frac{F}{A}$$
  • Where:
    • $\sigma$ = Pa (N \(m^{-2}\))
    • $F$ = N
    • $A$ = \(m^{2}\)
A is the cross-sectional area perpendicular to the force. For a circular wire, A = pi(d/2)^2.
Strain
$$\varepsilon = \frac{\Delta L}{L}$$
  • Where:
    • $\varepsilon$ = dimensionless
    • $\Delta L$ = m
    • $L$ = m
Strain has no units because it is a ratio of two lengths.
Young modulus
$$\begin{aligned} E &= \frac{\sigma}{\varepsilon} \\ &= \frac{FL}{A\Delta L} \end{aligned}$$
  • Where:
    • $E$ = Pa
A material property (independent of specimen dimensions). Gradient of the linear region of a stress-strain graph.
Springs in series
$$\frac{1}{k_{\text{total}}} = \frac{1}{k_1} + \frac{1}{k_2}$$
  • Where:
    • $k$ = N \(m^{-1}\)
Springs in series are less stiff (opposite to resistors in series). For n identical springs: k_total = k/n.
Springs in parallel
$$k_{\text{total}} = k_1 + k_2$$
  • Where:
    • $k$ = N \(m^{-1}\)
Springs in parallel are stiffer (opposite to resistors in parallel). For n identical springs: k_total = nk.