Hooke's Law /Stress-Strain Curve

Hooke's Law is widely used in science and engineering fields. In this article, a basic spring experiment will be introduced to understand Hooke's law and stress-strain curve for engineering applications.

 

 

Hooke's Law is used to describe an extension of an elastic object, such as a spring [1]. 

 

 

F = ke

Equation 1: The above equation states Hooke's law. 

 

 

• F : Force needed to stretch the spring (N)
• e : An extension (m)
• k : The spring constant, stiffness of the spring (N/m)

 

 

The experiment in figure 1 is used to find out a relationship between force and extension. When a weight is applied to a spring, the spring is pulled down. The greater force will pull the spring down further and increase the length of the spring from its original length. This change of length is called 'extension'. The relationship between the extension of the spring and applied force shows proportionality (figure 2). However, when the force reaches the limit of proportionality, the relationship is no longer proportional. Therefore, Hooke's law only applies before those elastic limits. 

 

 

Figure 1. The diagram describes the spring test to figure out the relationship between extension and force [1].

 

 

 

Figure 2. The graph shows the proportional relationship between force and extension on the spring [1].

 

 

 

 

Similarly, Hooke's law is extensively used to describe the relationship between strain (deformation) and stress.

 

From the tensile testing, applied stress and sample elongation are measured until the sample fractures and those recorded values are plotted to make the stress-strain curve (figure 3). From the test, many important material properties can be revealed such as Young's modulus, the ultimate tensile strength and the yield strength. Those parameters are critical for application design.

 

Be mindful, for engineering purposes, a cross-sectional area of materials does not change during the tension test. In reality, the actual area will decrease due to elastic and plastic deformation [3]. The engineering stress-strain curve is generally used in most cases unless stated as a true stress-strain curve which is based on the instantaneous cross-section area and length.   

 

 

 

Figure 3. The test piece is pulled further on the testing machine and tension is created on the piece [2].

 

 

 

 Figure 4 shows a typical engineering stress-strain diagram for ductile materials such as metals or structural steel. For brittle materials such as glass, aluminium, concrete, carbon fibre or diamond, the plastic region tends to be shorter or not show any plastic deformation (figure 5). In this article, we will look closely at the stress-strain curves of ductile materials. 

 

  

 

Figure 4. The typical stress-strain diagram for ductile materials, for instance, low carbon steel [4].

 

 

 

Figure 5. The graph shows different stress-strain curves between brittle materials and ductile materials [5].

 

 

 

The stress-strain curve can be divided into various sections that describe different behaviours (figure 4). The ratio of stress to strain within the proportional limit is called Young's modulus. It's a stiffness of the spring that describes as 'stiffness of a spring' in Hooke's law. When Young's modulus is small, a material can be stretched with small stress. The yield point is where the proportionality of stress to strain is lost, however, the elasticity of the material still exists. After the continuous deformation, there is a moment when the cross-sectional area gets narrow, so the amount of stress applied to the cross-sectional area gets smaller. This section is called 'Necking'. After the necking section, eventually, sample fracture will occur. 

 

 

Stress and Strain can be expressed like equation 2. 

 

Equation 2. The equations show axial stress and strain on ductile materials.

 

 

• σ : An axial stress (N/㎡ = Pa)
• F : Force (N)
• A0 : A bar of original cross-sectional Area (㎡)
• ε : Strain (unitless)
• L : Observed length of the bar
• L0 : The original length of the bar

 

 

 

 

 

 

 

 

[References]

[1] "Force on a spring - Hooke's law - CCEA - GCSE Combined Science Revision - CCEA Double Award - BBC Bitesize", BBC Bitesize. [Online]. Available: https://www.bbc.co.uk/bitesize/guides/z4fmkmn/revision/3. [Accessed: 20- Feb- 2022].

[2] A. Velling, Stress-Strain Curve. 2020.

[3] "Stress–strain curve - Wikipedia", En.wikipedia.org, 2022. [Online]. Available: https://en.wikipedia.org/wiki/Stress%E2%80%93strain_curve. [Accessed: 21- Feb- 2022].

[4] Attribution 4.0 International (CC BY 4.0), Typical stress vs. strain diagram for a ductile material (e.g. steel).. 2020.

[5] Stress–strain curve for brittle materials compared to ductile materials.. 2020.