Chapter 6

• Forces that produce extension are known as tensile forces and those that shorten an object are known as compressive forces.
• The force extension graph for a spring is linear to the elastic limit. In the linear region, the spring undergoes elastic deformation whereas after the elastic limit the spring undergoes plastic deformation
• Hooke's Law states that the extension of an object is directly proportional to the force applied as long as the elastic limit is not exceeded
• Hooke's law


  $F =$$kx$
• To investigate Hooke's Law, a spring is attached to a clamp stand and the initial extension is measured. Then increasing mass is added and the extension is measured for each. A force extension graph can then be plotted from this data.
• When an object undergoes elastic deformation the work done is fully recovered when the object returns to its original shape. If a material has undergone plastic deformation, then the work done is not fully recovered as this has been used to rearrange the atoms of the material into new permanent positions
• The area underneath a force-extension graph is equal to the work done.
• Work done on a spring is transferred to elastic potential energy within the spring.
• Elastic potential energy


  $E =$$\frac{1}{2} k x^2$
• Loading and unloading curve for metal wire.

  Beyond the elastic limit the wire is permanently extended after the force is removed.

  
• The loading curve for a rubber band.

  Rubber bands do not obey Hooke's Law. The rubber band will always return to its original length after the force is removed.

  The loop formed is a hysteresis loop.

  
• The loading curve for a polythene strip.


  A polythene strip does not obey Hooke's Law. They suffer plastic deformation under relatively little force.

  
• Tensile stress is the force applied per unit cross-sectional area. Measured in Pa


  $\sigma =$$\frac{F}{A}$
• Tensile strain is the extension per unit length. No units


  $\mathcal{E} =$$\frac{x}{L}$
• A stress-strain graph for mild steel wire


  From the origin to point P, the stress is directly proportional to strain where P is the limit of proportionality.

  E represents the elastic limit; elastic deformation occurs up to the elastic limit and plastic deformation beyond it.

  Y1 and Y2 are upper and lower yield points where the material extends rapidly. (typical of mild steel but uncommon in others)

  UTS is the ultimate tensile strength of the material, the maximum tensile stress the material can withstand when being stretched before it breaks.

  B is the breaking point of the material

  
• A ductile material can easily be drawn into a sheet
• A strong material is one with a high ultimate strength
• Youngs modulus is the ratio of stress to strain. Measured in Pa


  
  

  $E =$$\frac{\sigma}{\mathcal{E}}$
• The gradient of a stress-strain graph is the young's modulus.
• To determine the Youngs modulus of a wire, the diameter of a wire is measured using a micrometre. Then the wire is clamped to one end of a table along a ruler and a marker attached. This is used to measure the initial extension and any further change. Additional mass is added to the end of the wire that passes over a pulley and the extension is calculated. Then a stress-strain graph is plotted and the Youngs Modulus is the gradient of that line.
• Brittle materials show plastic behaviour up to their breaking point without elastic deformation.
• Polymeric materials can stretch a large amount before breaking, but rubber shows elastic behaviour whereas polythene shows plasic behaviour