Understanding True Stress and Engineering Stress in Tensile Testing

Tensile testing is a fundamental mechanical test where a sample is subjected to a controlled tension until failure. During this test, the material's response to stress and strain is measured. Two important concepts are engineering stress/strain and true stress/strain.

• Engineering Stress (\(\sigma_e\)): This is calculated by dividing the applied load (\(P\)) by the original cross-sectional area (\(A_0\)) of the sample. Mathematically, \(\sigma_e = \frac{P}{A_0}\).
• Engineering Strain (\(\epsilon_e\)): This is calculated by dividing the change in length (\(\Delta L\)) by the original length (\(L_0\)). Mathematically, \(\epsilon_e = \frac{\Delta L}{L_0}\).
• True Stress (\(\sigma_t\)): This is calculated by dividing the applied load (\(P\)) by the instantaneous cross-sectional area (\(A\)) at that moment. Mathematically, \(\sigma_t = \frac{P}{A}\).
• True Strain (\(\epsilon_t\)): This is calculated by integrating the infinitesimal changes in length divided by the instantaneous length. For a constant volume deformation, it can be related to engineering strain by \(\epsilon_t = \ln(1 + \epsilon_e)\).

Ultimate Tensile Stress (UTS)

The ultimate tensile stress (often called ultimate tensile strength) is the maximum stress value reached on the engineering stress-strain curve. It represents the maximum load that the material can withstand divided by the original cross-sectional area. After the material reaches its ultimate tensile stress, it typically begins to neck, especially in ductile materials.

Behaviour After Ultimate Tensile Stress

Necking is a phenomenon where the cross-sectional area of the sample starts to decrease significantly in a localized region. As necking occurs:

• The applied load required to continue deformation might decrease slightly due to the reduced area, leading to a drop in the engineering stress-strain curve after the ultimate tensile stress point.
• However, the actual stress supported by the material (the true stress) is calculated based on the instantaneous cross-sectional area, which is rapidly decreasing in the necked region.

True Stress at Fracture vs. Ultimate Stress

While the engineering stress may decrease after reaching the ultimate tensile stress (due to the decreasing load and constant original area in the calculation), the true stress continues to increase. This is because the reduction in the instantaneous cross-sectional area outweighs the change in load in the true stress calculation (\(\sigma_t = \frac{P}{A}\)). The material is supporting an increasing load per unit of actual area in the necked region.

Fracture eventually occurs in the necked region when the true stress reaches a critical value specific to the material under those conditions. Since the true stress accounts for the decreasing area and continues to rise after the ultimate tensile stress point until fracture, the true stress at fracture is always higher than the ultimate tensile stress (which is the peak of the engineering stress curve).

Therefore, during tensile testing, the true stress at fracture is higher than the ultimate stress.