I think , the failure of mild steel is essentially due to cup and cone formation unlike HYSD bars . and cup and cone failure happens coz of shear strain.
correct me If I am wrong .

Maximum shear strain energy theory

The most appropriate theory of failure for mild steel is the Maximum shear strain energy theory. This theory is also known as the Distortion Energy theory or Von Mises-Hencky theory.

Explanation:

Mild steel is a ductile material, which means it can undergo significant plastic deformation before failure. Therefore, the theory of failure for mild steel should be based on its ductile behavior rather than its brittle behavior.

The Maximum shear strain energy theory states that failure will occur when the total strain energy per unit volume in a material reaches a critical value. This critical value is related to the yield strength of the material.

Key points:

- The theory assumes that failure occurs when the distortion energy per unit volume reaches a critical value, regardless of the principal stresses acting on the material.

- It is based on the assumption that the total strain energy per unit volume is proportional to the total shear strain energy per unit volume.

- The theory takes into account both normal and shear stresses, but it gives more weightage to shear stresses.

- The theory is applicable to isotropic materials, such as mild steel, where the yield strength in tension and compression is the same.

Advantages of the Maximum shear strain energy theory:

1. The theory gives good correlation with experimental results for ductile materials like mild steel.

2. It provides a conservative estimate of failure, ensuring that the material is not overstressed.

3. The theory is relatively simple to apply and does not require the calculation of principal stresses or strains.

4. It is widely used in engineering design and analysis.

Limitations of the Maximum shear strain energy theory:

1. The theory is not applicable to materials that exhibit significant anisotropy or non-linear behavior.

2. It does not take into account the effects of hydrostatic stress, which can be important in some situations.

3. The theory assumes that failure occurs when the distortion energy reaches a critical value, but it does not provide any information about the mode or location of failure.

In conclusion, the Maximum shear strain energy theory is the most appropriate theory of failure for mild steel due to its ductile behavior and good correlation with experimental results.